Assaf, Barbara, Bohdan, Clément, Clément, Côme, Cyril Merci Arturo

Enzo

Fabio

Hiroshi, Houzhi, Junchao, Laure Guillaume

Luca

Marc Mi-Song Lucas

Quentin Rémy

Sothea, Sylvain, Thomas, William, Xiaoli, Yann, Yiyang Ziad Simon

Merci

Keywords: Produit de matrices aléatoires, Exposant de Lyapunov, Systèmes désordonnés, Comportement Product of random matrices, Lyapunov exponent, Disordered systems, Critical behaviour, Relevant disorder, Pinning model, Big jump, Path properties

Mes premiers et plus vifs remerciements vont à mon directeur Giambattista. Merci d'avoir accepté d'encadrer cette thèse. Tu as su partager ton enthousiasme et tes intuitions pour me guider dans ces recherches et les sujets variés et passionnants sur lesquels tu m'as proposé de travailler ont pour moi été une vraie chance. Ce fut un grand plaisir de travailler avec toi.

J'aimerais également remercier Philippe Carmona et Michail Loulakis pour leur relecture attentive du manuscrit et pour leurs nombreux commentaires et suggestions. Je suis honoré que Quentin Berger, Philippe Carmona, Francis Comets, Julien Poisat et Cristina Toninelli aient accepté de faire partie de mon jury de thèse. Quentin, j'ai apprécié nos discussions autour des modèles d'accrochage et je te remercie d'avoir su partager tes idées et ta connaissance de ce domaine avec enthousiasme.

Je remercie aussi Romain Dujardin pour des échanges autour d'un projet sur lequel, même s'il n'a pas pu aboutir, j'ai travaillé avec un grand plaisir mêlé de beaucoup de curiosité. J'adresse aussi ma reconnaissance à Nathalie et Valérie, pour leur grande disponibilité et leur investissement, qui permettent à la vie et au travail dans ce laboratoire d'être si agréables.

Merci également aux personnes qui m'ont donné le goût des mathématiques. Mes parents d'abord, mes professeurs et professeuses ensuite, de la primaire à l'université. La vie manque d'occasions de les remercier et même si je ne suis pas certain que chacun•e pourra lire ces lignes, je tiens à remercier particulièrement Christian Robesson, Isabelle Galotti, Christian Gilgenkrantz, Jean Feyler, et Bruno Arsac pour m'avoir formé aux mathématiques avec une grande passion, et enfin Zhan Shi pour m'avoir donné le goût de la théorie des probabilités. La force de votre engagement est un exemple pour moi.

J'adresse aussi de profonds remerciements à mes camarades doctorant•es du laboratoire qui ont su m'apporter un soutien permanent, sans lequel certains moments de la thèse auraient été difficiles à surmonter.

Liste des symboles et abréviations

IID : Indépendant•e•s et identiquement distribué•e•s N : Ensemble des entiers naturels {0, 1, 2, . . .} 1 A : Fonction indicatrice de l'ensemble A. On notera abusivement 1 f (x)∈A pour 1 f -1 (A) (x) P, E : Probabilité, espérance mathématique. Dans les chapitres 3 et suivants, ces notations sont réservées à l'espérance par rapport au renouvellement, et on utilise E et P pour l'espérance et la probabilité associées au désordre (ω j ).

x ∧ y : min(x, y) x ∨ y : max(x, y) x + : max(x, 0) log : Logarithme népérien (log(e) = 1) f (x) = o(g(x)) ou f (x) g(x) : f (x)/g(x) → 0 lorsque x → a f (x) = O(g(x)) : f (x)/g(x) est borné lorsque x → a f (x) ∼ g(x) : f (x) = g(x)(1 + o(1)) lorsque x → a , ∼ = : utilisés lorsqu'on ne souhaite pas être précis. P d-1 ou P d-1 (R) : Espace projectif : ensemble des droites de R d x : Un des deux vecteurs de norme 1 et de direction x ∈ P d-1

Convergence en probabilité : Lorsque µ n est une suite de mesures de probabilités, X n une suite de variable aléatoires réelles et c une constante réelle,

Motivation

Les produits de matrices aléatoires sont une structure récurrente dans l'étude de certains systèmes désordonnés, parmi lesquels on peut citer le modèle d'Anderson [START_REF] Bougerol | Products of random matrices with application to Schrödinger operators[END_REF][START_REF] Matsuda | Localization of Normal Modes and Energy Transport in the Disordered Harmonic Chain[END_REF] ou les chaînes harmoniques [START_REF] Dyson | The dynamics of a disordered linear chain[END_REF][START_REF] Schmidt | Disordered one-dimensional crystals[END_REF]. On s'intéresse ici à des versions désordonnées du modèle d'Ising. -Les atomes le constituant sont placés sur un réseau régulier ;

-Leur moment magnétique ne prend que deux états (+1 ou -1) ; -L'interaction entre atomes au sein du métal est réduite à une interaction entre atomes directement voisins (pas d'interaction à moyenne ou longue portée) et cette interaction est identique pour chaque atome ;

-Le champ externe est spatialement homogène.

Malgré cette série de simplifications, le modèle d'Ising homogène rend compte de la transition ferromagnétique/paramagnétique (typiquement la perte d'aimantation du fer lorsqu'il est chauffé au delà de la température de Curie).

Le fonctionnement d'un matériau ferromagnétique est bien plus complexe et on aimerait relâcher ces simplifications. Les recherches en physique théorique se sont notamment penchées sur des versions désordonnées du modèle d'Ising: l'inhomogénéité du matériau est modélisée par des variables aléatoires (e.g. des champs externes et/ou des coefficients de couplage aléatoires), le plus souvent supposées IID. On pourrait parler de modèle d'Ising en milieu aléatoire.

Nous nous penchons ici sur deux versions désordonnées très particulières du modèle d'Ising (au sens où la géométrie du désordre est spécifique) pour lesquels l'énergie libre associée s'exprime en terme d'un produit de matrices aléatoires. On se pose la question de l'influence de l'introduction du désordre sur l'énergie libre. Plus généralement on peut se demander si les propriétés du modèle à l'échelle macroscopique sont modifiées par l'introduction du désordre (on parle alors de désordre pertinent ou relevant) ou non (désordre non pertinent/irrelevant disorder ).

Chaine d'Ising plongée dans un champ aléatoire. Commençons par un modèle unidimensionnel. On se place sur le réseau torique Z/N Z, muni de coefficients de couplages J homogènes. En revanche le champ externe est inhomogène et on fait l'hypothèse que (h k ) k∈Z/N Z forme une famille IID de variables aléatoires intégrables. Posons = e -2βJ (quantité déterministe), et Z k = e -2βh k (variable aléatoire). Alors la fonction de partition s'écrit en termes d'un produit de matrices aléatoires IID :

Z N,β (ω) = e βN J+β N k=1 h k Tr(M 1, • • • M N, ), (1.1.3)
où (M k, ) est la famille de matrices aléatoires IID

M k, = 1 Z k Z k . (1.1.4)
Démonstration. Par définition

Z N,β = σ∈{-1,+1} Z/N Z exp β N k=1 h k σ k + β N k=1
Jσ k σ k+1 .

(1.1.5)

Introduisons la fonction définie sur {-1, +1} 2 par k (σ, τ ) = exp(β(h k σ + Jστ )), (1.1.6) qui correspond aux coefficients de la matrice de transfert (aléatoire) au site k :

L k = e β(h k +J) e β(h k -J) e β(-h k -J) e β(-h k +J) = e β(J+h k ) M k, . (1.1.7)
La fonction de partition se récrit

Z N,β = σ 1 ∈{-1,1}
. . . L'analyse de l'énergie libre du système se réduit ainsi à celle de la limite (si elle existe)

lim N →+∞ 1 N log Tr(M 1, • • • M N, ), M k, = 1 Z k Z k . (1.1.11)
L'étude de ce produit de matrices aléatoires est notamment l'objet de l'article de Derrida et Hilhorst [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF] (voir aussi [START_REF] Crisanti | Products of Random Matrices in Statistical Physics[END_REF] et les articles cités). On l'appellera donc modèle de , ces segments deviennent infiniment longs. On expliquera dans la section 1.3 que ces deux régimes correspondent à deux comportements asymptotiques de l'énergie libre bien différents.

Observons que la méthode des matrices transferts qui permet l'expression de l'énergie libre grâce à des produits de matrices (aléatoires) ne dépend pas de la forme du désordre et se généralise à d'autres versions désordonnées du modèle d'Ising unidimensionnel. D'autres études sont par exemple développées dans [START_REF] Crisanti | Products of Random Matrices in Statistical Physics[END_REF].

Modèle de McCoy-Wu. La structure de produits de matrices aléatoires émerge aussi pour des modèles bidimensionnels. Cela repose sur la célèbre méthode d'Onsager pour exprimer l'énergie libre du modèle d'Ising 2D à l'aide de matrices de transfert. Quoique cette dernière repose fortement sur les symétries du système, McCoy et Wu [START_REF] Mccoy | Theory of a Two-Dimensional Ising Model with Random Impurities. I. Thermodynamics[END_REF] proposent une version désordonnée du modèle d'Ising bidimensionnel, à champ nul (∀x, h x = 0), pour lequel la méthode d'Onsager reste valable. Il s'agit du modèle suivant. Les coefficients de couplage verticaux sont déterministes et prennent tous la même valeur J V . Les coefficients de couplage horizontaux sont aléatoires indépendants et de même loi selon chaque ligne, mais sont répétés sur toute la colonne correspondante (Figure 1.1). Soyons plus explicites. Donnons-nous un réel positif J V , et une suite (K i ) i∈Z de variable aléatoire positives IID. On pose : pour tout (i, j) ∈ Z 2 , J (i,j),(i,j+1) (ω) = J V J (i,j),(i+1,j) (ω) = K i (ω) .

(1.1.12) McCoy et Wu montrent que l'énergie libre du modèle s'écrit, à une fonction analytique près

f mw (β) := 1 4π π -π L mw β (θ)dθ, (1.1.16) où L mw β (θ) = lim N →+∞ 1 N log M β,N (θ) • • • M β,1 (θ) .
(1.1.17)

Les matrices M β,n (θ) apparaissant dans la résolution sont surprenamment similaires de celle du modèle de Derrida-Hilhorst. Ce dernier apparait ainsi comme un modèle jouet, ou au moins une première étape pour comprendre l'énergie libre f mw (β). Dans [START_REF] Mccoy | Theory of a Two-Dimensional Ising Model with Random Impurities. I. Thermodynamics[END_REF], l'analyse de L mw β (θ) dans une limite de désordre faible (ils considèrent une loi K 1 qui devient déterministe dans la limite N → +∞) les conduit à un modèle exactement soluble, pour lequel (1.1.16) devient possible à analyser. Ils prédisent que le désordre est pertinent et que la singularité est complètement lissée par le désordre : l'énergie f mw (β) présente une transition de phase d'ordre infini [START_REF] Fisher | Critical behavior of random transverse-field Ising spin chains[END_REF][START_REF] Iglói | Strong disorder RG approach-a short review of recent developments[END_REF] à une température inverse explicite β c . C'està-dire que f mw (β) est une fonction C ∞ de β mais présente une rupture d'analyticité en β c . Remarque 1.1.1. Shankar et Murthy [START_REF] Shankar | Nearest-neighbor frustrated random-bond model in d=2: Some exact results[END_REF] proposent une version légèrement différente: les interactions horizontales sont déterministes, et les interactions verticales sont aléatoires dans la direction horizontale, mais sont répétées sur chaque ligne. Quoique le système physique soit plus riches, en ce qu'il autorise la frustration, la même structure émerge.

Généralités sur l'exposant de Lyapunov

Dans cette section on expose, sans démonstration et sans rentrer dans un grand niveau de détail, une sélection de résultats de la théorie des produits de matrices aléatoires. Beaucoup de résultats généraux peuvent être trouvés dans le livre de Bougerol [START_REF] Bougerol | Products of random matrices with application to Schrödinger operators[END_REF] ou la vue d'ensemble de Viana [START_REF] Viana | Lectures on Lyapunov Exponents[END_REF].

Dans cette section, (M n ) n 0 désignera une suite IID de matrices aléatoires d × d, de loi commune notée µ. On notera sans distinction • la norme euclidienne sur R d et la norme qui lui est subordonnée sur M d (R).

Commençons par la définition de l'exposant de Lyapunov, qui est donnée par le théorème suivant.

Théorème 1.2.1 (Furstenberg-Kesten [START_REF] Furstenberg | Products of random matrices[END_REF]). Si E[log + M 1 ] < +∞, alors

1 n log M n • • • M 1 -→ n→+∞ L(µ) ∈ R ∪ {-∞}, (1.2.1)
presque sûrement et dans L 1 , où

L(µ) = inf n 1 1 n E log M n • • • M 1 (1.2.2)
est appelé exposant de Lyapunov, ou exposant caractéristique de la loi µ.

Exemple 1.2.2 (Matrices triangulaires, [START_REF] Bougerol | Products of random matrices with application to Schrödinger operators[END_REF]Ex. I.5.3]). Si les matrices M n sont presque sûrement triangulaires supérieures

M n (ω) =    λ (n) 1 (ω) ( * ) . . . (0) λ (n) d (ω)    , (1.2.3)
alors L(µ) = max

1 k d E[log |λ k |].
(1.2.4)

Exemple 1.2.3 (Matrice déterministe : Théorème de Gelfand). Lorsque µ = δ M , c'est-à-dire que les matrices M n sont déterministes, l'exposant de Lyapunov s'exprime simplement L(δ M ) = log(rayon spectral de M ).

(1.2.5)

En dehors de quelques exemples élémentaires, il est difficile d'obtenir une formule explicite de l'exposant de Lyapunov. On peut toutefois, avec des arguments de théorie erodique, en obtenir une expression implicite, en fonction de la mesure invariante d'une chaine de Markov.

Expression implicite et mesures invariantes

On suppose dans toute cette partie que µ ne charge que les matrices inversibles, autrement dit M ∈ GL d (R) presque sûrement. Le but de cette section est de donner des expressions implicites de l'exposant de Lyapunov, obtenues notamment par [START_REF] Furstenberg | Non-commuting Random Products[END_REF][START_REF] Furstenberg | Random matrix products and measures on projective spaces[END_REF][START_REF] Hennion | Loi des grands nombres et perturbations pour des produits réductibles de matrices aléatoires indépendantes[END_REF]. Elles reposent sur l'écriture « en somme téléscopique » suivante : pour tout y ∈ R d non nul,

1 N log M N . . . M 1 y = 1 N N n=1 log M n M n-1 . . . M 1 y M n-1 . . . M 1 y , (1.2.6)
qui se reformule comme moyenne de Birkhoff d'une chaine de Markov. Dans cette perspective, il est judicieux d'observer l'action naturelle de GL d (R) sur l'espace projectif

P d-1 = P(R d ) : pour toute matrice A ∈ GL d (R) et toute droite x = Ry ∈ P d-1 , A • (Ry) = R(Ay) (1.2.7)
Lorsque x ∈ P d-1 , on notera x un des vecteurs de norme 1 (indifféremment) de direction x. Il conviendra de ne pas confondre M • x et M x: le premier est un élément de l'espace projectif, tandis que le second est un vecteur de l'espace, dont c'est la norme qui nous intéressera. Enfin introduisons la notion de mesure invariante dans ce contexte.

Définition 1.2.4 (Mesure invariante). Une mesure de probabilité ν sur P d-1 est dite invariante par µ lorsque pour toute f ∈ C b (P d-1 ),

E[f (M • x)]dν(x) = f (x)dν(x).
(1.2.8)

Grâce à des théorèmes ergodiques usuels, l'écriture (1.2.6) fournit une première expression de l'exposant de Lyapunov. Théorème 1.2.5 (Furstenberg -Kifer [START_REF] Furstenberg | Random matrix products and measures on projective spaces[END_REF] et Hennion [START_REF] Hennion | Loi des grands nombres et perturbations pour des produits réductibles de matrices aléatoires indépendantes[END_REF]). Supposons que µ ne charge que les matrices inversibles, que E log + M < ∞ et que E log + M -1 < ∞. Dans ce cas L(µ) = sup

P d-1
E[log M x ]dν(x) : ν invariante par µ , (1.2.9)

et ce supremum est atteint.

Quoique le supremum soit atteint, l'expression (1.2.9) est peu aisément manipulable. On aimerait par exemple savoir pour quelle(s) mesure(s) ν il est atteint. Une première idée est de trouver des conditions pour que la mesure invariante soit unique. On cite souvent à ce propos un résultat de Furstenberg [START_REF] Furstenberg | Non-commuting Random Products[END_REF] (voir aussi [START_REF] Bougerol | Products of random matrices with application to Schrödinger operators[END_REF]Chap. 3]) qui démontre un critère d'unicité de la mesure invariante, sous des hypothèses dynamiques et géométriques sur le support de la loi µ. Malheureusement, nous nous concentrerons précisément sur des limites où ces hypothèses sont mises en défaut: en particulier, on travaillera avec des matrices aléatoires admettant plusieurs mesures invariantes. C'est pourquoi on ne reprend pas ce résultat ici. Au lieu de ça, on s'intéresse à un résultat de Furstenberg et Kifer, qui proposent plutôt une classe de mesures µ pour laquelle le supremum est atteint par n'importe quelle mesure ν. Introduisons à cet effet les deux définitions suivantes. Définition 1.2.6 (Espace invariant). Si µ est une loi sur M d (R), on dit que le sous espace L ⊂ R d est invariant sous l'action de µ si pour µ-presque toute matrice A, on a AL ⊂ L. Définition 1.2.7 (Irréductibilité). On dit que la probabilité µ sur M d (R) est irréductible si les seuls espaces invariants sous l'action de µ sont {0} et R d .

Théorème 1.2.8 ( [START_REF] Furstenberg | Random matrix products and measures on projective spaces[END_REF]). Si µ est une probabilité irréductible sur GL d (R) alors pour toute mesure invariante ν,

L(µ) = P d-1 E[log M x ]dν(x).
(1.2.10)

De plus, pour tout y ∈ R d \{0}, on a la convergence presque sûre

1 n log M n • • • M 1 y -→ n→+∞ L(µ).
(1.2.11)

Exemple 1.2.9. Soit, pour p ∈ (0, 1],

µ p = p δ M 1 +(1-p)δ M 2 , avec M 1 = 2 0 0 1 2 et M 2 = 0 -1 1 0 . (1.2.12)
Observons que la probabilité ν = 1 2 δ (0,1) + 1 2 δ (1,0) est toujours une mesure invariante. Par ailleurs, lorsque p ∈ (0, 1], la mesure µ p est irréductible. Ainsi, grâce au Théorème 1.2.8,

L(µ p ) = P 1 (p log M 1 x + (1 -p) log M 2 x ) dν(x) = 0.
(1.2.13) Remarque 1.2.10. En général, toutes les mesures invariantes ne maximisent pas l'intégrale dans (1.2.9). L'hypothèse d'irréductibilité est cruciale, et quasiment nécessaire. Par exemple, pour la matrice déterministe diagonale M = Diag [START_REF] Alexander | The effect of disorder on polymer depinning transitions[END_REF][START_REF] Alexander | Equality of critical points for polymer depinning transitions with loop exponent one[END_REF], les mesures δ (1,0) et δ (0,1) sont invariantes, mais seule la seconde donne l'exposant de Lyapunov. Les matrices diagonales, et en général les matrices aléatoires ayant des sous-espaces invariants non triviaux, sont pathologiques dans la théorie développée par Furstenberg et al. Ce sont précisément des perturbations de matrices diagonales qui vont nous intéresser dans les sections suivantes, et c'est la raison pour laquelle on s'attend à des comportements singuliers.

On ne développe pas davantage la théorie générale. Ces seuls résultats nous seront suffisants. De nombreux travaux récents en théorie ergodique s'intéressent aux propriétés de l'exposant de Lyapunov (analyticité, harmonicité, etc.) ou de la mesure invariante (existence d'une densité et régularité de celle-ci). Il est à noter que ce sont des questions difficiles même pour des lois aussi élémentaires que 1 2 (δ A + δ B ) (A, B désignant deux matrices déterministes données).

Dans les deux dernières sous-sections sont exposés quelques résultats de régularité de l'exposant de Lyapunov, en tant que fonction de µ.

Continuité de l'exposant de Lyapunov

Dans la perspective d'étudier des énergies libres définies par un exposant de Lyapunov, la question de la régularité de celui-ci par rapport à la loi de la matrice est cruciale. Commençons par nous intéresser à la continuité de l'application µ → L(µ), où l'espace des mesures est muni de la distance de la convergence étroite. Signalons que la continuité n'est pas toujours satisfaite, même pour des lois élémentaires. L'exemple suivant l'illustre.

Exemple 1.2.11 (Exemple de discontinuité). Reprenons l'exemple 1.2.9. Clairement p → µ p est continue sur [0, 1] au sens de la convergence étroite. Mais L(µ p ) = 0 pour p ∈ (0, 1], tandis que L(µ 0 ) = log 2. Ainsi p → L(µ p ) n'est pas continue en 0.

Toutefois on peut trouver, pour une suite de mesures µ k convergeant étroitement vers µ, des critères sous lesquels L(µ k ) converge vers L(µ). Il convient déjà d'imposer une condition d'équiintégrabilité. Définition 1.2.12. On dit qu'une suite de distributions (µ k ) sur GL d (R) est équiintégrable lorsque

sup k 0 log + A 1 log + A >T dµ k (A) + log + A -1 1 log + A -1 >T dµ k (A) -→ T →∞ 0.
(1.2.14) Proposition 1.2.13 (Kifer [START_REF] Kifer | Perturbation of random matrix products[END_REF]). Donnons-nous une suite équiintégrable (µ k ) k 0 de probabilités sur GL d (R) convergeant étroitement vers une probabilité µ. Alors

lim sup k→∞ L(µ k ) L(µ).
(1.2.15)

Si de plus µ est irréductible, alors

L(µ k ) -→ k→∞ L(µ). (1.2.16)
Remarque 1.2.14. Le cas de mesures irréductibles n'est pas « trop restrictif » : Kifer a montré [START_REF] Kifer | Perturbation of random matrix products[END_REF] que l'ensemble des mesures irréductibles est un ouvert dense de l'ensemble des probabilités sur GL d (R).

Certes, en un sens il y a beaucoup de mesures irréductibles ; toutefois, dans l'étude de systèmes physiques, notamment des systèmes désordonnés, les lois considérées sont souvent des perturbations d'une matrice déterministe (ou d'une matrice diagonale pour les modèles de Derrida -Hilhorst et McCoy -Wu), qui n'est en particulier que rarement irréductible. Pour ce qui concerne les perturbations d'une matrice déterministe, un critère simple existe. Théorème 1.2. 15 ([72]). Soit M ∈ GL d (R). Il existe R > 0 tel que pour toute suite (µ k ) k 0 de probabilités à support dans la boule {A ∈ GL d (R) : A -M R} et convergeant étroitement vers δ M , on a la convergence des exposants caractéristiques :

L(µ k ) -→ k→∞ L(δ M ).

Dérivabilité et Analyticité

Pour les modèles auquel on s'intéressera, la continuité par rapport aux paramètres sera presque immédiatement acquise. En revanche son analyticité, ou même sa dérivabilité, sera un problème difficile, et pourtant crucial pour comprendre les transitions de phases des modèles physiques concernés. Même dans un cadre général, de tels résultats sont rares. Nous n'en citerons qu'un, qui aura toutefois des conséquences puissantes sur nos modèles. Il s'agit du théorème de Ruelle [START_REF] Ruelle | Analycity properties of the characteristic exponents of random matrix products[END_REF], repris par Dubois [START_REF] Dubois | Real cone contractions and analyticity properties of the characteristic exponents[END_REF], que l'on énonce dans un cadre simplifié. Donnons d'abord une définition de la réelle-analyticité. Définition 1.2.16. Si V est un espace de Banach, une application f :

V → R est dite analytique réelle au voisinage du point x 0 ∈ V s'il existe une suite (a k ) -où pour chaque k, a k est une forme k-linéaire continue sur V -, ainsi qu'un réel R > 0 vérifiant k≥0 a k R k < ∞, et tels que pour tout x ∈ B(0, R), f (x 0 + x) = k≥0 a k (x, x, . . . , x).
(1.2.17)

On introduit également la notion de cône, utilisée dans le théorème de Ruelle :

M (C 2 ) ⊂ C 1 ∪ (-C 1 ) : (1.2.18) le cône C 2 est contracté par M .
Pour une matrice aléatoire M quelconque, définie sur un espace probabilisé Ω, on note, lorsqu'il existe, L(M ) l'exposant de Lyapunov associé. On note aussi E(M ) l'espace de Banach des matrices aléatoires A définies sur Ω telles que

A E(M ) := sup ω∈Ω A(ω) M (ω) < +∞, ( 1
x k V k (1.2.20)
est une fonction analytique de x au voisinage de 0.

Signalons, pour ce qui concerne la dérivabilité, les travaux de Hennion [START_REF]Dérivabilité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes à coefficients positifs[END_REF]. Il établit que pour des matrices à coefficients positifs et k + 2 fois dérivables, l'exposant de Lyapunov correspondant est k fois dérivable. Entre autres hypothèses techniques, ce résultat requiert que les coefficients de la matrice aléatoire soient, avec un probabilité non nulle, strictement positifs simultanément. Malheureusement, cette dernière restriction l'empêche de fournir des résultats satisfaisants dans les modèles qui nous intéressent. En fait, pour ces modèles physiques, les seuls cas où les travaux de Hennion [START_REF]Dérivabilité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes à coefficients positifs[END_REF] pourraient s'appliquer, le théorème de Ruelle s'y applique déjà, fournissant un résultat plus fort.

Le Modèle de Derrida-Hilhorst

Revenons au modèle de Derrida-Hilhorst, introduit dans la section 1.1, et pour lequel les matrices aléatoires sont

M k, = 1 Z k Z k . (1.3.1)
On notera M une matrice générique ayant la même loi. Le Théorème 1.2.1 garantit l'existence de la limite

L Z ( ) = lim N →+∞ 1 N log M N, • • • M 1, (1.3.2)
pourvu que E[log + Z] soit finie. Le choix de remplacer la trace de la formule par la norme ne change pas la limite (voir le Lemme 2.2.3). On souhaite comprendre la régularité de → L Z ( ). Avant d'aborder le coeur de l'étude, commençons par deux réductions simples. On constate, en factorisant Z dans la matrice M , que

L Z ( ) = E[log Z] + L 1/Z ( ). (1.3.3)
Par conséquent on supposera toujours, sans perte de généralité, que E[log Z] 0. Dans ce cas L Z (0) = 0. De même, une conjugaison par la matrice diagonale Diag(-1, 1) donne

L Z ( ) = L Z (-). (1.3.4)
La fonction L Z ( ) est paire et l'étude peut donc être restreinte à > 0.

Lorsque Z a un support compact dans (0, +∞), observant que pour > 0, le cone {(y 1 , y 2 ) | y 1 0, y 2 0} est contracté par l'action de M , le Théorème 1. L'étude centrale de [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF] concerne le cas où

E[log Z] < 0 et E[Z] > 1. (Figure 1.2c) Dans ce cas, par convexité de la fonction β → E[Z β ], il existe un unique α ∈ (0, 1) tel que E[Z α ] = 1. Derrida et Hilhorst prédisent que L Z ( ) ∼ C Z 2α , (1.3.5)
pour une certaine constante C Z > 0 semi-explicite. Il convient de noter que puisque la fonction → L Z ( ) est paire (1.3.4), seule les puissances paires de sont non-singulières en 0. Ainsi le comportement (1.3.5) est singulier même lorsque α est un demi-entier.

Que se passe-t-il dans les autres cas : E[Z] 1 ou E[log Z] = 0 ? La littérature physique fournit des prédictions pour ceux-ci, dont l'exposition est facilitée par un paramétrage par l'exposant α, que nous allons décrire. On se place désormais dans le cas où P(Z > 1) > 0, et pour faciliter l'exposition des conjectures et résultats, on supposera que Z est une variable aléatoire bornée.

-Lorsque E[log Z] < 0, grâce à la convexité de la fonction 1.2b) Les mêmes auteurs donnent un argument [35, §3] prédisant que L Z ( ) a un développement en puissances de 2 , où le coefficient devant 2n est fini si et seulement si

β → E[Z β ], il existe un unique α > 0 tel que E[Z α ] = 1. (Figure
E[Z n ] < 1, c'est-à-dire si et seulement si n < α.
Par analogie, avec le cas α ∈ (0, 1) (1.3.5), il est probable que, lorsque α > 1, ce soit un terme 2α qui brise l'analyticité. Ils laissent toutefois ces prédictions au rang de conjectures.

-Lorsque E[log Z] = 0, β = 0 est la seule solution de E[Z β ] = 1 et on convient de poser α = 0 (Figure 1.2a). Pour un désordre Z très spécifique entrant dans ce cadre, Nieuwenhuizen et Luck [START_REF] Nieuwenhuizen | Exactly soluble random field Ising models in one dimension[END_REF] estiment l'exposant de Lyapunov et prévoient Nous résumons ces prédictions dans la conjecture suivante. [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF] proposent un calcul explicite de la fonction périodique C Z (log ) pour un choix très spécifique (et singulier) de la loi Z.

L Z ( ) ∼ C Z log(1/ ) . (1.3.6) β E[Z β ] 0 1 α = 0 (a) E[log Z] = 0. α 1 β E[Z β ] 0 (b) E[log Z] < 0. α 1 β E[Z β ] E[Z] 0 1 (c) E[log Z] < 0 and E[Z] > 1.
L( ) = c 1 2 + • • • + c α 2 α + C Z 2α + o( 2α ); (1.3.7) 2. Si α ∈ {1, 2, . . .}, L( ) = c 1 2 + • • • + c α-1 2(α-1) + C Z 2α log + o( 2α log ); (1.3.8) 3. Si α = 0, L( ) = C Z log(1/ ) + o (log ) -1 . ( 1
Plusieurs articles se sont penchés sur cette conjecture ces dernières années, apportant une démonstration rigoureuse à une partie de ces prédictions. Genovese et al. [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF] proposent une démonstration rigoureuse de la prédiction (1.3.5) de Derrida et Hilhorst lorsque α ∈ (0, 1) (c'est le résultat principal de [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF]). Observons que la singularité 2α est dans ce cas la partie dominante de L( ). Nous nous sommes penchés sur le cas α 1, pour lequel nous obtenons [START_REF] Havret | Regular Expansion for the characteristic exponent of a product of 2 × 2 random matrices[END_REF] (repris dans le chapitre 2) le développement (1.3.7) jusqu'à la singularité 2α exclue. En parallèle, Comets et al. [START_REF] Comets | Continuum limit of random matrix products in statistical mechanics of disordered systems[END_REF] se sont penchés sur une limite en désordre faible du modèle de Derrida-Hilhorst. Dans cette limite, le modèle est exactement soluble et l'exposant de Lyapunov correspondant admet une expression explicite. De façon surprenante, le développement (1.3.7) persiste dans la limite de désordre faible, singularité incluse (Sous-section 1.4.2). De plus, ce modèle s'inscrit dans un cadre de perturbation de la matrice identité qui avait déjà suscité de l'intérêt indépendamment (Sous-section 1.4.1) ; et surtout, il offre un éclairage nouveau sur l'approche de McCoy et Wu. Nous y reviendrons dans les sections suivantes. Pour l'heure, recentrons-nous sur le modèle de Derrida-Hilhorst à proprement parler et détaillons les méthodes de [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF][START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF][START_REF] Havret | Regular Expansion for the characteristic exponent of a product of 2 × 2 random matrices[END_REF].

Ces études reposent sur l'expression semi-explicite (1.2.10) de l'exposant de Lyapunov, qui dans ce contexte peut s'écrire (voir Chapitre 2)

L( ) = E log(1 + 2 X ) = +∞ 0 log(1 + 2 x)dν (x), (1.3.10)
où X désigne une variable aléatoire de loi stationnaire, notée ν , pour la chaîne de Markov sur R +

x n+1 = Z n+1 1 + x n 1 + 2 x n . (1.3.11) En d'autres termes X (d) = Z 1 + X 1 + 2 X . (1.3.12)
Vue la formule (1.3.10), toute l'inconnue réside dans la mesure ν , probabilité invariante d'une suite homographique aléatoire. Elle n'admet aucune expression explicite, et malgré l'apparente simplicité de la chaine de Markov, comprendre, même grossièrement, la mesure invariante ν est malaisé. D'ailleurs, lorsque α 1, une compréhension suffisamment fine de celle-ci fait encore défaut, raison pour laquelle la singularité 2α échappe à l'étude (voir le paragraphe 1.3.2.3).

Développement « régulier » en puissances de 2

Détaillons un argument heuristique justifiant la prédiction de B. Derrida et H. Hilhorst pour α > 1. Ce n'est pas précisément l'argument développé dans [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF], quoiqu'il s'en rapproche, mais il fournit le même niveau de prédiction. Celui que l'on développe ici repose presque entièrement sur la simplification de la chaine (1.3.11) par la chaine de Markov linéaire

x n+1 = Z n+1 (1 + x n ), (1.3.13)
dont on note ν 0 l'unique mesure invariante sur R + , et X 0 la variable aléatoire correspondante. Notre argument repose en grande partie sur le fait que ν 0 constitue une bonne approximation de ν , suffisante au moins pour obtenir le développement (1.3.7) jusqu'à l'ordre α . Contrairement à ν , la mesure invariante ν 0 admet une expression explicite :

X 0 (d) = +∞ n=1 Z 1 • • • Z n . (1.3.14)
Malgré l'apparente simplicité de la chaine, son étude est riche et son analyse peu aisée. L'étude de la queue de X 0 a notamment suscité de l'intérêt (voir notamment [START_REF] Kesten | Random difference equations and Renewal theory for products of random matrices[END_REF], ou [START_REF] Kevei | A note on the Kesten-Grincevičius-Goldie theorem[END_REF][START_REF]Implicit renewal theory in the arithmetic case[END_REF] pour un état de l'art récent). Ces auteurs établissent, sous des hypothèses techniques que l'on omet ici, que

P(X 0 x) x→∞ ∼ c ν x -α . (1.3.15)
En fait un comportement aussi précis de la queue ne sera utile que plus tard et pour l'heure, on peut se contenter de lire sur la formule (1.3.14) que, pour β > 0, le moment E[X β 0 ] est fini si et seulement si E[Z β ] < 1. Cette observation permet déjà de comprendre, au point de vue heuristique, pourquoi s'attendre au développement (1.3.7) prévu par [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF]. Pour cela, utilisons le développement du logarithme dans l'expression (1.3.10). Pour 0 n < β n + 1, écrivons formellement

L Z ( ) = E 2 X - 1 2 4 X 2 + . . . + (-1) n+1 n 2n X n + O( 2β X β ) = 2 E[X ] - 1 2 4 E[X 2 ] + . . . + (-1) n+1 n 2n E[X n ] + O( 2β E[X β ]) 2 E[X ] - 1 2 4 E[X 2 ] + . . . + (-1) n+1 n 2n E[X n ] + O( 2β E[X β 0 ]).
(1. Ces trois lignes de calcul peuvent aisément être rendues rigoureuses, en se contentant de trouver des bornes adéquates pour les moments E[X k ] (voir chapitre 2). Un pas reste à franchir : les coefficients (-1)

k+1 k 2k E[X k ], k < α ne sont pas des constantes, et l'approximation E[X n ] E[X k
0 ] est trop grossière. Pour parvenir au développement (1.3.7), il faut exhiber un développement en puissances de 2 de chacun des moments E[X k ], k < α. À cette fin, utilisons l'identité (1.3.12), et de sa conséquence

E[X k ] = E[Z k ]E (1 + X ) k (1 + 2 X ) k , (1.3.17)
que l'on peut développer, comme dans (1.3.16), en puissance de 2 . Une utilisation de ces identités permet d'obtenir, à travers une procédure récursive, un développement en puissance de 2 de plus en plus précis des moments E[X k ], que l'on peut ultimement écrire

E[X k ] = c n,0 + c n,1 2 + c n,2 4 + • • • + O( 2β-2k E[X β ]). (1.3.18)
On peut noter que la puissance β = α est de nouveau un obstacle pour ce développement. Cependant, s'il est naturel de postuler que le terme suivant est 2α , rien pour l'instant ne vient le conforter. Cette analyse (menée de manière rigoureuse dans le chapitre 2) aboutit à une version faible de la conjecture (1.3.7) :

L( ) = c 1 2 + • • • + c α 2 α + R( ), où R( ) = o( 2 α ). (1.3.19)
On aimerait un équivalent précis R( ) ∼ C Z 2α , que ne fournit pas notre approche. Avançons-nous toutefois un petit peu et voyons où la méthode qui précède faut. L'obtention de ce terme singulier oblige à se confronter aux moments non bornés de X , et donc avec sa queue. C'est ici que l'approche précédente est trop faible, car l'approximation X X 0 sur laquelle elle repose devient fort imprécise pour estimer la queue de X . Plus précisément, l'approximation

1 + x 1 + 2 x 1 + x (1.3.
[Z] > 1 et E[log Z] < 0, pour lequel α ∈ (0, 1) et la singularité 2α devient le comportement dominant de L Z ( ). Leur prédiction prend la forme L Z ( ) 0 ∼ C Z 2α . (1.3.21)
Pour parvenir à une telle prédiction, Derrida et Hilhorst analysent la loi stationnaire ν et en donne une forme très précise, reposant sur une étude à deux échelles. À l'échelle standard, l'approximation ν ν 0 vaut. À l'échelle -2 , une autre approche est nécessaire car la chaine de Markov changée d'échelle n'a pas, dans la limite → 0, de probabilité invariante non triviale. La loi ν y est approchée par une troncature d'une mesure invariante de masse infinie. Un recollement approprié de ces deux échelles fournit selon eux une bonne approximation de la loi invariante ν . Cet aperçu grossier étant brossé, entrons dans les détails de leur analyse.

Dans un premier temps, Derrida et Hilhorst font le constat de ce qui nous a déjà servi auparavant : lorsque → 0, la mesure invariante ν de la chaine

x n+1 = Z n+1 1 + x n 1 + 2 x n (1.3.22)
converge vers ν 0 , l'unique probabilité invariante de la chaine

x n+1 = Z n+1 (1 + x n ). (1.3.23)
L'approximation ν ν 0 sera utilisée sur cette échelle (régime I). Ils regardent ensuite la mesure ν après avoir « dézoomé » d'un facteur -2 (régime II), c'est-à-dire la mesure invariante ω de la chaine de Markov

σ n+1 := 2 x n+1 = Z n+1 2 + σ n 1 + σ n , (1.3.24)
Dans la limite 0, cette chaine devient

σ n+1 = Z n+1 σ n 1 + σ n . (1.3.25)
En observant que σ n Z n • • • Z 1 σ 0 , on se heurte à un premier constat qui peut laisser pessimiste : dans le cadre où E[log Z] < 0, l'unique probabilité invariante de la chaine (1.3.25) est la mesure de Dirac δ 0 . Toutefois, le point essentiel est que cette chaine possède aussi des mesures invariantes (positives) de masse infinie. Celle que Derrida et Hilhorst proposent de choisir peut être construite comme suit. Prenons une mesure invariante ω de la chaine (1.3.24) choisie de telle sorte que ω ([1, +∞)) = 1. On peut montrer qu'elle admet une limite (au sens faible), notée ω 0 . Cette mesure ω 0 est une mesure invariante (de masse infinie) de la chaine (1.3.25). Ils justifient qu'elle doit satisfaire

ω 0 ([σ, +∞)) σ→0 ∼ c ω σ -α . (1.3.26)
La prédiction de Derrida et Hilhorst est que cette mesure ω 0 décrit, à une renormalisation près, le comportement de la queue de X . Ils prédisent en fait que la mesure ν peut-être décrite assez précisément par le recollement à deux échelles, de la mesure ν 0 et de la mesure ω 0 :

ν ([x, +∞)) = P(X x) ∼ = P(X 0 x) pour x -1 a( )ω 0 ([ 2 x, +∞)) pour x -1
(1.

L Z ( ) = +∞ 0 log(1 + 2 x)dν (x) ∼ = -1 0 log(1 + 2 x)dν 0 (x) + a( ) +∞ -1 log(1 + σ)dω 0 (σ) = C Z 2α + O( 1+α ), avec C Z = c ν c ω +∞ 0 log(1 + σ)dω 0 (σ).
(1.3.29)

Remarquons que, lorsque α ∈ (0, 1), c'est la queue de X qui donne la contribution majoritaire à L Z ( ). Nommément, a( ) Les prédictions de Derrida et Hilhorst s'avèrent exactes et Genovese, Giacomin et Greenblatt [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF] proposent une démonstration mathématique de (1.3.29) sous la forme suivante.

+∞ -1 log(1 + σ)dω 0 (σ) ∼ C Z 2α , (1.3.30) tandis que -1 0 log(1 + 2 x)dν 0 (x) = O( 1+α ) = o( 2α
Théorème 1.3.4 (Genovese -Giacomin -Greenblatt [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF]). On suppose

1. E[log Z] < 0, et E[Z] > 1 ;
2. La variable aléatoire Z est à densité par rapport à la mesure de Lebesgue, et cette densité est

C 1 et à support compact [c -, c + ] ⊂ (0, +∞). Alors il existe κ > 0 et C Z > 0 tels que L( ) = C Z 2α + O( 2α+κ ). (1.3.32)
Suivant fortement la voie tracée par Derrida et Hilhorst, le travail majeur de [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF] est de proposer un version rigoureuse de l'approximation (1.3.27) de la mesure invariante. Sans entrer dans les détails techniques, donnons les grandes lignes de leur approche. La construction de ν 0 et ω 0 ne pose pas de problème et relève de techniques standards. De même que l'estimation de leur queue respective (1.3.15) et (1.3.26), reposant sur l'analyse de transformées de Mellin. Ils proposent ensuite de considérer la mesure, donnée par la fonction de répartition (voir la figure dans [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF])

γ ([x, +∞)) := ν 0 ([x, +∞)) pour x -1 a( )ω 0 ([ 2 x, +∞)) pour x -1 , (1.3.33)
où a( ) est choisi de tel sorte que que la fonction de répartition est continue en -1 . Comme on l'a déjà observé (1.3.29), cette mesure donne le résultat souhaité :

+∞ 0 log(1 + 2 x)dγ (x) = C Z 2α + O( 1+α ). (1.3.34)
Mais il reste à établir que γ est proche de ν , dans un sens suffisamment fort pour garantir (1.3.32). Une des idées principales des auteurs de [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF] est d'introduire des normes à poids sur les mesures signées η :

η β = +∞ 0 x β-1 |η((x, +∞))|dx, (1.3.35)
paramétrées par β ∈ (0, α), qui vont permettre de mesurer la proximité entre ν et γ . Ces normes présentent un double avantage. D'une part elles suffisent à contrôler l'exposant de Lyapunov :

+∞ 0 log(1 + 2 x)dν 1 (x) - +∞ 0 log(1 + 2 x)dν 2 (x) 2β ν 1 -ν 2 β . ( 1
f (x)dT η(x) = +∞ 0 E f Z 1 + x 1 + 2 x dη(x). (1.3.37)
Grosso modo, T donne la loi transformée après un pas de la chaine (1.3.22). En particulier la mesure invariante ν en est un point fixe T ν = ν . La propriété de contraction s'écrit

T η 1 -T η 2 β c β η 1 -η 2 β , c β ∈ (0, 1), (1.3.38)
et par suite

ν -γ β 1 1 -c β T γ -γ β . (1.3.39)
Cette astuce standard sur les opérateurs contractants permet de mesurer la proximité de ν et γ sans connaitre la mesure invariante ν . Ainsi, pour montrer que γ est proche de la vraie mesure invariante, il suffit de vérifier qu'elle est elle-même « presque invariante », c'est-à-dire que T γγ β tend suffisamment vite vers 0, ce que les auteurs établissent au terme d'un long calcul technique. Ils obtiennent , pour un certain δ > 0,

T γ -γ β = O( 2α+β ) + O( α+β+δ ), (1.3.40)
ce qui, moyennant un choix approprié de β, conclut leur preuve.

Observons que cette approche n'est pas sans lien avec celle de la sous-section 1.3.1 qui permit d'obtenir la partie régulière du développement. Les normes • β sont en effet intimement liées au contrôle des moments : lorsque η est la loi d'une variable aléatoire positive Y , on a L'approche de Genovese et al. n'a permis d'aboutir à la singularité 2 α que lorsque α ∈ (0, 1), c'est-à-dire lorsque celle-ce est la partie prépondérante de L Z ( ). Voyons pourquoi les tentatives de production d'un résultat analogue pour α > 1 échouent. Rappelons la conclusion de la sous-section 1.3.1 :

η β = E[Y β ], et E[Y β ] -E[X β ] η -ν β . ( 1
L( ) = c 1 2 + • • • + c α 2 α + R( ), où R( ) = o( 2 α ). (1.3.42)
Pour faciliter l'exposition, on reste dans le cadre α ∈ N. Afin d'estimer précisément le terme R( ), dont on suppose qu'il doit être C Z 2α , on pourrait être tenté d'élargir l'heuristique proposée par Derrida et Hilhorst au cas α > 1. En se souvenant que pour obtenir la partie régulière, on avait besoin de prendre en compte tous les moments de X , l'idée pourrait être d'essayer de montrer

ν ([x, +∞)) = P(X x) ∼ = P(X x) pour x q a( )ω 0 ([ 2 x, +∞)) pour x q , ( 1
P(X c -2 ) c P(X 0 c -2 ). (1.3.44)
Malheureusement, aucun argument ne permet à ma connaissance de justifier cette dernière estimation. À ce titre il faut constater le fait remarquable que dans [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF], la justification de l'approximation ν ([x, +∞)) ≈ a( )ω 0 ([ 2 x, +∞)) (1.3.45) pour x -1 n'est jamais faite individuellement. Certes on peut déduire de leur contrôle de T γγ que, pour α ∈ (0, 1),

+∞ -1 x β-1 a( )ω 0 (( 2 x, +∞)) -ν ((x, +∞)) dx = O( 2α-β ).
(1.3.46)

Mais cette estimation repose fortement sur la construction totale de γ , comprenant les deux échelles. C'est une telle construction qui fait précisément défaut pour α > 1.

À défaut de l'équivalent désiré R( ) ∼ C Z 2α , un raffinement de la méthode présentée dans la sous-section 1.3.1 fournit une estimation qualitative de R( ). En effet les erreurs commises lors des développements successifs en puissance de 2 (voir (1.3.16)

M η = I 2 + ηU + η 2 K, (1.4.1)
où K est une matrice déterministe et U une matrice aléatoire dont les coefficients sont centrés. Ils prédisent que l'exposant de Lyapunov correspondant s'écrit

L(η) = η 2 λ + O(η 3 ). ( 1
M η = I 2 + n k=1 η k V k + η n+1 W η , avec E[V 1 ] = (0). ( 1
f dν η = n-2 k=0 η k f ρ k d + O(η n-1
).

(1.4.4)

Remarque 1.4.2. En particulier ν η -→ ρ 0 d , au sens de la convergence étroite, lorsque η → 0.

Vues la formule intégrale (1.2.10), on obtient naturellement le comportement de L(η) lorsque η → 0.

Théorème 1.4.3. Si pour tout η = 0 assez petit, la matrice ηV 1 + η 2 K est irréductible, alors il existe des coefficients (λ k ) k 2 , tels que

L(η) = n k=2 λ k η k + O(η n+1 ).
(1.4.5)

Nous proposons en Annexe 1.A une démonstration de ces résultats, motivée par la restriction des hypothèses des théorèmes.

Remarque 1.4.4. En réalité, les développements à l'ordre n -2 des mesures invariantes et à l'ordre n de l'exposant de Lyapunov exigent uniquement que les queues des coefficients des matrices aléatoires V k et W η soient suffisamment contrôlées pour que le développement (1.A.5) du Lemme 1.A.1 ait lieu. Typiquement, pour les matrices aléatoires V k , des queues exponentielles suffisent.

Peut-on espérer l'analyticité ? Lorsque, par exemple,

M η = I 2 + +∞ k=1 η k V k , ( 1
V 1 = 0 -W W 0 , V 1 = 0 0 0 W ou V 1 = 0 W 0 0 , ( 1
y n+1 = 1 Z n Z n y n = y n + 0 Z n Z n -1 y n . (1.4.8)
Formellement, on fixe = 0, α ∈ R, η > 0, et des variables gaussiennes standards (N n ) n 0 et on pose

Z η (n) = exp σηN n -α σ 2 2 η 2 .
(1.4.9)

Il convient de constater que E[(Z η ) α ] = 1.

(1.4.10)

On considère alors la chaîne de Markov donnée par X η (0) = X(0) = 0, et

X η (n + 1) = X η (n) + 0 η 2 η 2 Z η (n + 1) Z η (n + 1) -1 X η (n) = M η (n)X η (n). (1.4.11)
On s'intéresse toujours à l'exposant de Lyapunov associé

L(η) = lim n→+∞ 1 n log X η (n) = lim n→+∞ 1 n log M η (n) • • • M η (1)
.

(1.4.12)

Observons que, dans la limite η → 0, la matrice aléatoire M η admet le développement en puissances de η :

M η = I 2 + ηV 1 + η 2 V 2 + O(η 3 ), (1.4.13)
où

V 1 = 0 0 0 σN , K = E[V 2 ] = 0 σ 2 2 (1 -α) . ( 1
L σ,α ( ) = σ 2 4 x K α-1 (x) K α (x) , où x = 4 σ 2 .
(1.4.18)

Ils retrouvent ainsi l'expression de λ 2 obtenue par Comtet et al. [START_REF] Comtet | The Lyapunov exponent of products 2×2 matrices close to the identity[END_REF] (voir le paragraphe consacré dans la sous-section 1.4.1). Au vu de la conjecture 1.3.2, le résultat le plus remarquable est le suivant.

Théorème 1.4.5. Il existe des fonctions rationnelles c j (α), j 1, telles que les développement suivants soient vérifiés, lorsque tend vers 0.

1. Si α ∈ (0, +∞)\N, 4 σ 2 L σ,α ( ) = c 1 x 2 + . . . + c α x 2 α + 2 Γ(1 -α) 4 α Γ(α) x 2α + O(x min(2 α ,4α) ). (1.4.19) 2. Si α ∈ {1, 2, . . .}, 4 σ 2 L σ,α ( ) = c 1 x 2 + . . . + c α-1 x 2(α-1) + 2 (-1) α 4 α (α -1)! 2 x 2α log x + O(x 2α ). (1.4.20) 3. Si α = 0, 4 σ 2 L σ,α ( ) = 1 log(1/x) + O (log(1/x)) -2 .
(1.4.21)

4. Si α < 0, 4 σ 2 L σ,α ( ) = 2|α| + 4 σ 2 L σ,|α| ( ). (1.4.22)
Il est assez frappant que dans la limite continue, qui est une limite de désordre faible, le comportement de l'exposant de Lyapunov coïncide parfaitement, à tous les ordres, avec celui prévu par la conjecture 1. Les deux prochaines sous-sections (1.A.1 et 1.A.2) reprennent l'approche de Sadel et Schulz-Baldes pour aboutir à la démonstration de ces deux théorèmes. On omettra volontairement dans cette première phase la preuve de trois lemmes techniques. Ce sont ces trois lemmes dont la démonstration dépend des détails du modèle et pour lesquels on justifie (sous-section 1.A.3) que les hypothèses de [START_REF] Sadel | Random Lie group actions on compact manifolds: a perturbative analysis[END_REF] peuvent être amoindries dans le contexte que l'on considère.

1.A.1 Aperçu de la méthode

Considérons l'opérateur

M η : C ∞ (P 1 ) -→ C ∞ (P 1 ) (1.A.1) défini par M η f (x) = E[f (M η • x)]. (1.A.2)
On souhaite établir que toute mesure invariante ν η peut s'écrire

dν η ρ η d , ρ η = n-2 k=0 η k ρ k + O(η n-1 ), (1.
M η = I + n k=2 η k M k + O(η n+1 ). (1.A.4)
Ce développement est rigoureux dans le sens suivant.

Lemme 1.A.1. Il existe une famille d'opérateurs différentiels M k : C ∞ (P 1 ) → C ∞ (P 1 ), k n tels que pour toute fonction f ∈ C ∞ (P 1 ),

M η f = f + n k=2 η k M k f + O(η n+1 ). (1.A.5)
Le O, dans cette identité et dans toute les suivantes, est uniforme en x ∈ P 1 .

La démonstration est immédiate (il s'agit de faire un développement à l'ordre n de E[f (M η • x)]) et repose sur le fait que les variables V k sont bornées. On ne la détaille pas ici.

L'identité M * η ρ η = ρ η devient, pourvu qu'une permutation des sommes soit possible,

ρ η = I + n k=2 η k M * k ρ η + O(η n+1 ) = ρ η + n k=2 n-2 l=0 η k+l M * k ρ l + O(η n+1 ) = ρ η + n m=2 η m m k=2 M * k ρ m-k + O(η n+1 ). (1.A.6)
Ainsi nous obtenons le système d'équations

m k=2 M * k ρ m-k = 0, 2 m n. (1.A.7)
On peut le récrire sous forme récursive : On aura besoin pour la démonstration de considérer l'opérateur adjoint de M 2 au sens L 2 . Cela exigerait habituellement de prolonger M 2 . Ici, vue la forme de l'opérateur, on peut se contenter de travailler avec des opérateurs agissant sur C ∞ (P 1 ). On introduit à ce propos les notions d'opérateur différentiel et d'adjoint d'un tel opérateur. Définition 1.A.2 (Opérateur Différentiel -Adjoint). On appellera opérateur différentiel sur C ∞ (P 1 ) tout opérateur s'écrivant K = k i=0 α i D i , où D est l'opérateur de dérivation et les α i sont des applications C ∞ sur P 1 . On appellera adjoint de K et on notera K * l'opérateur différentiel sur C ∞ (P 1 ) donné, pour f ∈ C ∞ (P 1 ), par la formule

M * 2 ρ 0 = 0, (1.A.8) et pour tout 1 m n -2, M * 2 ρ m = - m k=1 M * k+2 ρ m-k . ( 1
K * f = k i=0 (-1) i D i [α i f ]. Lemme 1.A.3. Il existe une fonction positive ρ 0 ∈ C ∞ (P 1 ), satisfaisant ρ 0 d = 1, et telle que Ker M * 2 = Rρ 0 . (1.A.11) Lemme 1.A.4. Pour tout k 3, Im M * k ⊂ {f : f d = 0} = Im M * 2 .
Ces deux lemmes garantissent déjà l'existence d'une suite de fonctions ρ k ∈ C ∞ (P 1 ) satisfaisant (1.A.8), (1.A.9) et (1.A.10). La preuve du Théorème 1.4.1 nécessite un dernier lemme technique.

Lemme 1.A.5. (Ker M * 2 ) ⊥ ⊂ Im M 2 .
Avant de démontrer ces trois lemmes, voyons comment ils permettent de démontrer le Théorème 1.4.1.

Démonstration du Théorème 1.4.1. Soit ν η une mesure invariante. Démontrons par récurrence que pour -1 r n -2,

f dν η = r k=0 η k f ρ k d + O(η r+1 ).
(1.A.12)

Le résultat est immédiat pour r = -1. Fixons 0 r n -2 et supposons que pour toute fonction g ∈ C ∞ (P 1 ), on ait

gdν η = r-1 k=0 η k gρ k d + O(η r ) (1.A.13) Soit f ∈ C ∞ (P 1
). Grâce au Lemme 1.A.5, on peut décomposer

f = f, ρ 0 + M 2 h = f ρ 0 d + M 2 h, (1.A.14) pour une certaine fonction h ∈ C ∞ (P 1 ). Ainsi f dν η = f ρ 0 d + M 2 hdν η . ( 1
hdν η = M η hdν η = hdν η + r+2 k=2 η k M k hdν η + O(η r+3 ). (1.A.16) Donc M 2 hdν η = - r k=3 η k-2 M k hdν η + O(η r+1 ).
(1.A.17)

L'hypothèse de récurrence assure que

M 2 hdν η = - r k=3 η k-2 r-k+2 j=0 η j M k hρ j d + O(η r+1 ), (1.A.18)
soit, en réarrangeant les sommes (m

:= j + k -2, i := k -2), M 2 hdν η = - r m=1 η m m i=1 M i+2 hρ m-i d + O(η r+1 ) = - r m=1 η m m i=1 hM * i+2 ρ m-i d + O(η r+1 ) = - r m=1 η m h × m i=1 M * i+2 ρ m-i d + O(η r+1 ). (1.A.19)
Avec la définition (1.A.9) des fonctions ρ m , on obtient

M 2 h dν η = r m=1 η m h (M * 2 ρ m ) d + O(η r+1 ) = r m=1 η m (M 2 h)ρ m d + O(η r+1 ).
(1.A.20)

Chapter 1. Produits de matrices aléatoires : Introduction

Puisque ρ m d = 0 pour tout m 1, on a (M 2 h)ρ m d = (f -f, ρ 0 )ρ m d = f ρ m d . (1.A.21)
Et donc

M 2 hdν η = r m=1 η m f ρ m d + O(η r+1 ). ( 1
L(η) = P 1 E [log M η x ] dν η (x). (1.A.23)
Rappelons que x désigne un des deux vecteurs de norme 1 et de direction x. D'après le Lemme 1.A.1, il existe des fonctions H p ∈ C ∞ (P 1 ) telles que, uniformément en x,

E [log M η x ] = n p=0 η p H p (x) + O(η n+1 ). (1.A.24) En fait H 0 est nulle car M 0 = I 2 , et H 1 aussi car E[V 1 ] = (0). Ainsi L(η) = n p=2 η p P 1 H p (x)dν η (x) + O(η n+1 ). (1.A.25)
Le résultat découle alors immédiatement du Théorème 1.4.1.

1.A.3 Étude de l'opérateur différentiel M 2 et preuves des lemmes

Il s'agit de démontrer les lemmes 1.A.3, 1.A.4 et 1.A.5, sous l'hypothèse que la matrice ηV 1 + η 2 K est irréductible. On commence par donner une expression de l'opérateur différentiel M 2 . Dorénavant, on utilise la paramétrisation

P 1 = - π 2 , π 2 - π 2 ∼ π 2 (1.A.26) Écrivons V 1 = u 1 u 2 u 3 u 4 , K = δ 1 δ 2 δ 3 δ 4 . (1.A.27) Posons ensuite T = tan x, puis ω 1 (x) = u 3 + u 4 T -u 1 T -u 2 T 2 , ω 2 (x) = δ 3 + δ 4 T -δ 1 T -δ 2 T 2 + (u 1 + u 2 T ) 2 T -(u 1 + u 2 T )(u 3 + u 4 T ). (1.A.28) Alors, notant b 1 (x) = E ω 1 (x) 2 2(1 + T 2 ) 2 , b 2 (x) = E ω 2 (x)(1 + T 2 ) -T ω 1 (x) 2 (1 + T 2 ) 2 , (1.A.29) on a M 2 f (x) = b 1 (x)f (x) + b 2 (x)f (x). (1.A.30)
Ainsi M 2 est un opérateur différentiel, et son adjoint s'écrit

M * 2 f (x) = (b 1 f ) (x) -(b 2 f ) (x). (1.A.31)
Passons aux démonstrations des trois lemmes. Les preuves vont dépendre de la forme des matrices V 1 et K. Dans l'article [START_REF] Sadel | Random Lie group actions on compact manifolds: a perturbative analysis[END_REF], C. Sadel, et H. Schulz-Baldes ne traitent que le cas où la matrice aléatoire V 1 est irréductible. En réalité, leur démonstration peut être adaptée sous des hypothèses un peu moins restrictives. Commençons par examiner le champ des possibles. On observe rapidement que, à une conjugaison déterministe près, on est forcément dans l'un des cas suivants

1. V 1 est irréductible; 2. u 3 = 0, t (1, 0) est la seule direction invariante, et P(u 1 = u 4 ) < 1; 3. u 3 = 0, t (1, 0) est la seule direction invariante, et P(u 1 = u 4 ) = 1; 4. u 2 = u 3 = 0 et P(u 1 = u 4 ) < 1; 5. V 1 = uI 2 , où u est une variable aléatoire réelle.
On traite séparément ces cinq cas, à l'exception du 3 e . Les méthodes sont similaires mais des difficultés techniques apparaissent dans le traitement des 2 e et 4 e cas, qui méritent d'être mentionnées.

Preuve lorsque V 1 est une homothétie. Ce dernier cas est sans doute le plus facile. Puisque ηV 1 +η 2 K est irréductible, c'est que K doit être irréductible et, quitte à conjuguer, on peut supposer que

K = a -b b a , b > 0. (1.A.32) Alors les fonctions b 1 et b 2 sont constantes, b 1 = 0, b 2 = b et M 2 f = bf . La
T ∈ R ∪ {±∞}, ω 1 (x) 1 + T 2 = u 3 + u 4 T -u 1 T -u 2 T 2 1 + T 2 (1.A.33)
ne peut pas être presque sûrement nul, sinon la droite t (1, T )

(ou t (0, 1) lorsque T = ∞) serait stable par V 1 , contredisant l'irréductibilité. Par conséquent pour tout x ∈ P 1 , b 1 (x) = 1 2 E ω 1 (x) 1 + T 2 2 = 0. (1.A.34)
Démonstration du Lemme 1.A.3. Il s'agit de résoudre

(b 1 ρ 0 ) -(b 2 ρ 0 ) = 0. (1.A.35)
Après une première intégration on obtient, pour un certain α ∈ R,

(b 1 ρ 0 ) -b 2 ρ 0 = -α. (1.A.36)
Soit φ une primitive de b 2 /b 1 , choisie pour être C ∞ sur P 1 \{0} (il peut arriver que φ(0 -) = φ(0 + )). Quitte à changer x en -x, disons φ(0 + ) φ(0 -). On a, sur P 1 \{0},

(b 1 ρ 0 exp(-φ)) = -α exp(-φ) (1.A.37) Ainsi ρ 0 (x) = 1 b 1 (x) exp(φ(x)) -α x 0 + exp(-φ(t))d (t) + β (1.A.38) Avec cette formule, ρ 0 est C ∞ sur P 1 \{0}. Puis les constantes α et β sont choisies de sorte que ρ 0 (0 -) = ρ 0 (0 + ). (1.A.39) et P 1 ρ 0 d = 1. (1.A.40)
Pour être explicite, la condition (1.A.39) requiert

α = β P 1 e -φ -1 1 -e φ(0 + )-φ(0 -) , (1.A.41) c'est-à-dire ρ 0 (x) = β b 1 (x) exp(φ(x)) 1 - x 0 + exp(-φ(t))d (t) P 1 exp(-φ)d 1 -e φ(0 + )-φ(0 -) (1.A.42)
On vérifie ensuite aisément que ρ 0 est partout strictement positive, et que, grâce à (1.A.36), elle est C ∞ sur P 1 . Puis la constante β est choisie pour satisfaire (1.A.40).

Démonstration du Lemme

1.A.5. Soit f ∈ C ∞ (P 1 ) telle que f, ρ 0 = 0. On souhaite résoudre b 1 h + b 2 h = f. (1.A.43)
En multipliant l'équation par ρ 0 , puis en utilisant (1.A.35) et (1.A.36), on obtient la forme équivalente

f ρ 0 = b 1 ρ 0 h + b 2 ρ 0 h = (b 1 ρ 0 h) -(b 2 ρ 0 h) + 2αh . (1.A.44) Soit F une primitive de f ρ 0 , qui existe et est C ∞ sur P 1 car f, ρ 0 = 0. L'équation peut alors être récrite F + C = (b 1 ρ 0 h) -b 2 ρ 0 h + 2αh. (1.A.45) Soit ψ une primitive de c = b 2 /b 1 -2α/(b 1 ρ 0 ) sur P 1 \{0}. L'équation devient (F + C) exp(-ψ) = (b 1 ρ 0 h exp(-ψ)) . (1.A.46)
On souhaite définir

h(x) = 1 ρ 0 (x)b 1 (x) e ψ(x) x 0 + (F (t) + C)e -ψ(t) d (t) + D . (1.A.47)
C'est une solution de l'équation (1.A.45) sur P 1 \{0}. Les constantes C et D doivent être choisies de sorte que h(0 -) = h(0 + ). Alors l'équation (1.A.45) assure que h ∈ C ∞ (P 1 ). Il suffit de prendre D arbitraire puis

C = 1 P 1 e -ψ d D e ψ(0 + )-ψ(0 -) -1 - P 1 F e -ψ d . (1.A.48)
On omet la démonstration du Lemme 1.A.4, qui suit le même schéma.

Preuve lorsque V 1 a une unique direction stable. Les autres cas sont plus techniques mais les méthodes de preuve restent approximativement les mêmes. C'est pourquoi on démontre le Lemme 1.A.3 uniquement. On suppose maintenant que u 3 = 0, que t (1, 0) est la seule direction invariante, et que P(u 1 = u 4 ) < 1. Dans ce cas la fonction b 1 est strictement positive sur P 1 \{0} et a un zéro d'ordre deux au point 0. Par ailleurs, puisque la matrice aléatoire ηV 1 + η 2 K est irréductible, on sait que δ 3 = 0 et, quitte à conjuguer encore, on peut supposer δ 3 > 0.

Démonstration du Lemme 1.A.3. On souhaite résoudre

(b 1 ρ 0 ) -b 2 ρ 0 = -α. (1.A.49) Soit φ une primitive de b 2 /b 1 qui soit C ∞ sur P 1 \{0}. On a alors (b 1 ρ 0 exp(-φ)) = -α exp(-φ). (1.A.50) Donc ρ 0 (x) = 1 b 1 (x) exp(φ(x)) -α exp(-φ(t))dt + β . (1.A.51) Soit a, b ∈ R tels que b 2 (x) b 1 (x) = a x 2 + b x + O x→0 (1). (1.A.52) Le choix δ 3 > 0 garantit que a > 0. Ainsi φ(x) → +∞ lorsque x → 0 -et donc on est forcé de prendre ρ 0 (x) = α b 1 (x) exp(φ(x)) 0 x exp(-φ(t))dt, (1.A.53) avec la convention 0 x = π/2 x + 0 -π/2 (1.A.54) si x ∈ (0, π/2]. On définit ainsi une fonction C ∞ et strictement positive sur P 1 \{0}. On peut vérifier que lorsque x → 0 ± , ρ 0 (x) = α b 2 (0) -α b 2 (0) b 2 (0) 2 x + o(x), (1.
(b 1 ρ 0 ) -b 2 ρ 0 = -α. (1.A.56) Soit φ (resp. ϕ) une primitive de b 2 /b 1 qui soit C ∞ sur (0, π/2) (resp. sur (-π/2, 0)). On a alors, pour x ∈ (0, π/2), (b 1 ρ 0 exp(-φ)) = -α exp(-φ). (1.A.57) Donc ρ 0 (x) = 1 b 1 (x) exp(φ(x)) -α exp(-φ(t))dt + β . (1.A.58) Et la même chose sur (-π/2, 0), avec ϕ et d'autres constantes α, β. Soit a, b ∈ R tels que b 2 (x) b 1 (x) = a x 2 + b x + O x→0 (1).
(1.A.59)

Le choix δ 3 > 0 garantit que a > 0. Ainsi φ(x) → -∞ lorsque x → 0 + . De même φ(x) → -∞ lorsque x → (π/2) -, et ϕ(x) → +∞ lorsque x → 0 -et lorsque x → (-π/2) + . Ainsi l'unique solution est ρ 0 (x) = β b 1 (x) exp(φ(x))1 x∈(0,π/2) . (1.A.60)
On définit ainsi une fonction C ∞ et positive sur P 1 \{0, π/2}. On vérifie aisément qu'en fait ρ 0 est C ∞ sur P 1 et satisfait l'équation (1.A.56) avec α = 0. On ajuste enfin la constante β de sorte que ρ 0 d = 1.

Démonstration du Lemme 1.A.3 lorsque δ 2 < 0. Cette fois φ(x) → -∞ lorsque x → 0 + , et φ(x) → +∞ lorsque x → (π/2) -. Et symétriquement ϕ(x) → +∞ lorsque x → 0 -et ϕ(x) → -∞ lorsque x → (-π/2) + . Ainsi l'unique solution est donnée par ρ 0 (x) = α b 1 (x) exp(φ(x)) π/2 x exp(-φ(t))dt si x ∈ [0, π/2] α b 1 (x) exp(ϕ(x)) 0 x exp(-ϕ(t))dt si x ∈ [-π/2, 0] (1.A.61)
On vérifie la validité des raccords en 0 et π/2 et qu'on définit ainsi une fonction C ∞ sur P 1 et satisfaisant l'équation (1.A.56). On ajuste enfin la constante α de sorte que ρ 0 d = 1.

1.A.4 Questions ouvertes et Commentaires

Que dire lorsque ηV 1 + η 2 K n'est pas irréductible ? La première chose à noter est que la méthode développée précédemment échoue : on ne peut même pas trouver de solution intégrable ρ 0 à l'équation M * 2 ρ 0 = 0. Pour autant, la fonction η → L(η) est-elle singulière en 0 ? Commençons par le cas caricatural où la matrice M η = I 2 + ηV 1 + η 2 K est presque sûrement triangulaire supérieure. Dans ce cas

L(η) = max E log(1 + ηu 1 + η 2 δ 1 ), E log(1 + ηu 4 + η 2 δ 4 ) = max η 2 2 (2δ 1 -u 2 1 ) + η 3 3 u 3 1 + O(η 4 ), η 2 2 (2δ 4 -u 2 4 ) + η 3 3 u 3 4 + O(η 4 ) . (1.A.62) Ainsi L(η) présente une singularité en 0 lorsque 2δ 1 -u 2 1 = 2δ 4 -u 2 4 et u 3 1 = u 3 4 : L(η) = η 2 2 (2δ 1 -u 2 1 ) + 1 3 max η 3 u 3 1 , η 3 u 3 4 + O(η 4 ) = η 2 2 (2δ 1 -u 2 1 ) + η 3 6 ( u 3 1 + u 3 1 ) - 1 6 |η| 3 | u 3 1 -u 3 4 | + O(η 4 ).
(1.A.63)

Se posent alors les questions suivantes, pour lesquelles nous ne connaissons pas d'argument permettant d'y répondre.

-Cette singularité subsiste-t-elle lorsque la partie principale ηV 1 +η 2 K est triangulaire, mais la matrice M η est irréductible ?

-Si oui, peut-on néanmoins toujours écrire

L(η) = λη 2 + O(η 3 ) ? Que dire si E[V 1
] n'est pas nulle ? Un autre cas qui n'est pas couvert par [START_REF] Sadel | Random Lie group actions on compact manifolds: a perturbative analysis[END_REF], celui où la matrice V 1 n'est pas centrée. Si toutefois la matrice

K := E[V 1 ] est irréductible alors, quitte à la conjuguer supposons que K = a -b b a . (1.A.64) Dans ce cas M η = I + ηM 1 + • • • et M 1 = bf et le Théorème 1.4.1 reste vrai, avec ρ 0 = 1.
Finalement on obtient un développement de la forme Abstract We consider a product of 2 × 2 random matrices which appears in the physics literature in the analysis of some 1D disordered models. These matrices depend on a parameter > 0 and on a positive random variable Z. Derrida and Hilhorst (J Phys A 16:2641, 1983, §3) conjecture that the corresponding characteristic exponent has a regular expansion with respect to up to -and not further -an order determined by the distribution of Z. We give a rigorous proof of that statement. We also study the singular term which breaks that expansion.

L(η) = aη + n-1 k=2 λ k η k + O(η n ). (1.A.65) En revanche, si E[V 1 ] n'

Introduction

Random matrix products appeared in the physics literature as a powerful tool to study disordered systems, ranging from Anderson model [START_REF] Bougerol | Products of random matrices with application to Schrödinger operators[END_REF][START_REF] Matsuda | Localization of Normal Modes and Energy Transport in the Disordered Harmonic Chain[END_REF] to disordered harmonic chains 35 [START_REF] Dyson | The dynamics of a disordered linear chain[END_REF][START_REF] Schmidt | Disordered one-dimensional crystals[END_REF] or disordered Ising model (discussed below). Among that wide range of models, the present work focuses on a very specific one, introduced by B. Derrida and H. Hilhorst in [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF] to study the strong interaction limit of a 1D disordered Ising model.

Let (Z n ) be iid non-negative and non-deterministic random variables, with law µ. For > 0, consider the matrices

M n, = 1 Z n Z n . (2.1.1)
We will write Z for a random variable with law µ and M for the associated matrix. In fact, we will use Z instead of µ to formulate our assumptions and results. The (leading) Lyapunov exponent -also called characteristic exponent -is the growth rate of their product:

L( ) = L Z ( ) = lim n→+∞ 1 n log M n, • • • M 1, . (2.1.2)
We will be particularly interested in the behaviour of L( ) in the limit → 0. 2×2 matrices of the form (2.1.1) have appeared several times to express the free energy of the disordered 1D Ising model [START_REF] Crisanti | Products of Random Matrices in Statistical Physics[END_REF][START_REF] De Calan | On the distribution of a random variable occurring in 1D disordered systems[END_REF][START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF][START_REF] Nieuwenhuizen | Exactly soluble random field Ising models in one dimension[END_REF], where the limit → 0 represents a regim of very strong interactions. It is also used in the celebrated work by B. McCoy and T. T. Wu [START_REF] Mccoy | Theory of a Two-Dimensional Ising Model with Random Impurities. I. Thermodynamics[END_REF] to study a 2D Ising model with 1D disorder, as well as in a similar model proposed by R. Shankar and G. Murthy [START_REF] Shankar | Nearest-neighbor frustrated random-bond model in d=2: Some exact results[END_REF] which includes frustrated interactions.

From a mathematical point of view, a wide literature proposed to study these models and more general matrix products. One should cite the seminal work by H. Furstenberg et al. [START_REF] Furstenberg | Non-commuting Random Products[END_REF][START_REF] Furstenberg | Products of random matrices[END_REF] and Oseledec's theorem [START_REF] Oseledec | A multiplicative ergodic theorem: Lyapunov characteristic exponents for dynamical systems[END_REF] (see [START_REF] Viana | Lectures on Lyapunov Exponents[END_REF] for a review). Looking at our own task, Furstenberg-Kesten theorem [START_REF] Furstenberg | Products of random matrices[END_REF] asserts that the limit (2.1.2) exists almost surely and is deterministic, as long as E[log + M ] is finite (here E[log + Z] < +∞ suffices). When vanishes, the matrix M n, tends to a diagonal matrix and the Lyapunov exponent can be explicitly computed thanks to the law of large numbers: L(0) = max(0, E[log Z]). However, diagonal matrices are a degenerate case in the theory developed by H. Furstenberg et al. and one expects, in most cases, that → L( ) is singular around 0.

It is worth stressing from now that the diagonal matrix M n,0 is still random. Therefore we are not in the framework of weak disorder limits such as [START_REF] Campanino | Anomalies in the one-dimensional Anderson model at weak disorder[END_REF][START_REF] Sadel | Random Lie group actions on compact manifolds: a perturbative analysis[END_REF][START_REF] Zanon | Weak disorder expansion of Liapunov exponents in a degenerate case[END_REF] in which the matrix M n,0 is deterministic. The main reference paper for our analysis is rather [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF].

General Conjecture and known results

The present work is motivated by the recent mathematical progress by Genovese et al. [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF]. In this paper some physical predictions, about the limiting behaviour of L( ) when vanishes, mainly stated in [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF] (see also [START_REF] Crisanti | Products of Random Matrices in Statistical Physics[END_REF][START_REF] De Calan | On the distribution of a random variable occurring in 1D disordered systems[END_REF][START_REF] Nieuwenhuizen | Exactly soluble random field Ising models in one dimension[END_REF]), are proven. However the physical predictions go beyond. We now first formulate these predictions in the form of conjectures, which detail the expected limiting behaviour of L( ), depending on the distribution of Z.

Then we explain what has been proven and what our contribution is.

Definition 2.1.1. A real-valued random variable ξ is said to be arithmetic when there exists a constant c > 0 such that c ξ ∈ Z ∪ {±∞} almost surely.

Conjecture 2.1.2. Assume that log Z is nonarithmetic.

1. Suppose in addition that E[log Z] < 0 and that there exists α ∈ (0, +∞) such that

E[Z α ] = 1.
-If α ∈ {1, 2, . . .}, then, as goes to 0,

L( ) = α k=1 (-1) k+1 k 2k + (-1) α +1 C Z 2α + o( 2α ), (2.1.3)
where, for k α , k is a positive rational function of

E[Z], . . . , E[Z k ]; and C Z is a positive real number. -If α ∈ {1, 2, . . .}, then L( ) = α-1 k=1 (-1) k+1 k 2k + (-1) α+1 C Z 2α log(1/ ) + o 2α log , (2.1.4)
where the coefficients ( k ) are the same positive rational functions of Z's moments as before; and C Z is still a positive constant.

2. If E[log Z] = 0 -it is the "α = 0" case -then L( ) = C Z log(1/ ) + o (log 1/ ) -1 . (2.1.5)
The same references motivate further comments.

Remark 2.1.3. The coefficients ( k ) appearing in the conjecture can be computed recursively. For instance

1 = E[Z] 1 -E[Z] , 2 = (1 + E[Z]) 2 E[Z 2 ] + 2E[Z] 2 (1 -E[Z 2 ]) 2(1 -E[Z]) 2 (1 -E[Z 2 ]) . (2.1.6)
Precise recursive formulas will be derived in Section 2.3. However it is not clear that a simple closed formula for k can be derived. By contrast, apart from a few special situations, the calculation of the constant C Z is a very hard problem [28, §4.2.1]. On another note, in all the instances developed in Conjecture 2.1.2, the constant C Z should be replaced by a multiplicatively periodic function of if log Z is arithmetic. A precise computation of such a multiplicatively periodic function C Z is made in [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF] for a very specific (and singular) distribution of Z.

Remark 2.1.4. We discuss in this remark the instances which are excluded by the conjecture. The conjecture actually covers almost all the cases where E[log Z] 0 and P(Z > 1) > 0, except the one discussed in the item 5 of Remark 2.1.7.

The case E[log

Z] > 0 (it corresponds to α < 0) boils down to E[log Z] < 0 by factorizing Z in the matrix M : L Z ( ) = E[log Z]+L 1/Z ( ).
Similarly, by conjugating by the matrix Diag(-1, 1), one observes that L is an even function: L( ) = L(-).

It implies that the behaviour 2α is rather | | 2α , so it is actually singular even when α is a half-integer.

2. If Z 1 almost surely (that is "α = +∞"), then L( ) admits a regular expansion with respect to 2 up to any order. Is it smooth or analytic in a neighborhood of 0? The problem is still open, except if Z ∈ [0, 1η] almost surely, for some η ∈ (0, 1). If so then it is a consequence of a result by D. Ruelle [START_REF] Ruelle | Analycity properties of the characteristic exponents of random matrix products[END_REF] that L( ) is a real analytic function of around 0.

3. That same theorem of D. Ruelle also ensures that the Lyapunov exponent L( ) is always an analytic function of on (0, +∞).

Very little of Conjecture 2.1.2 has been made mathematically rigorous. To our knowledge only (2.1.3) has been successfully tackled [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF], and only for α ∈ (0, 1). When α ∈ (0, 1), that is E[log Z] < 0 and E[Z] > 1, the singularity 2α happens to be the leading behaviour of L( ) and (2.1.3) takes the form:

L( ) 0 = C Z 2α + o( 2α ). (2.1.7)
This simplifies in a substantial way the analysis: Derrida and Hilhorst in [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF] (see also [START_REF] De Calan | On the distribution of a random variable occurring in 1D disordered systems[END_REF][START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF][START_REF] Nieuwenhuizen | Exactly soluble random field Ising models in one dimension[END_REF]) explicitly give, based on a two scale argument, a probability measure that is expected to be close, when is small, to the invariant probability for the action of M on the projective space P 1 (R) (that is, the distribution of X , in the next paragraph's notations). Then, they use this probability to compute the Lyapunov exponent. This two scale analysis is made rigourous by G. Genovese et al. [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF], who show that this probability measure is indeed close to the invariant measure in a suitable norm, and this control is sufficiently strong to yield precisely (2.1.7). It appears to be rather challenging to follow the same steps for α 1: the guess for the invariant probability would have to be tuned to yield the α terms of the regular expansion and the singular 2α term. Even at a heuristic level, such a construction is lacking. Note, in particular, that in [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF], the α 1 case is treated in a expedite way, without reference to the invariant probability, and without capturing the singularity 2α .

On the other hand, a weak disorder limit of the model has been investigated. In this limit, the product of random matrices becomes a stochastic differential equation system. An exactly solvable structure emerges from that SDE and the analog of (2.1.3) and (2.1.4) has been shown to hold (see [START_REF] Grabsch | One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers[END_REF] for the case α ∈ (0, 2) and [START_REF] Comets | Continuum limit of random matrix products in statistical mechanics of disordered systems[END_REF] for the general case). As pointed out in [START_REF] Comets | Continuum limit of random matrix products in statistical mechanics of disordered systems[END_REF], it is rather remarkable that the structure of (2.1.3) and (2.1.4) holds also in the weak disorder limit and this appears to be a rather deep fact. Nonetheless, the fact that the conjecture holds in the weak disorder limit is far from being a mathematical proof of the conjecture for products of matrices.

The main aim of our work is to approach (2.1.3) and (2.1.4). Our results are the following.

1. L( ) admits a regular expansion in powers of 2 , up to order 2 α , or 2(α-1) in the integer case (that is the regular part of (2.1.3) and (2.1.4)).

2. We prove that the next order term after this regular part, call it R( ), satisfies, as goes to 0, for instance in the non-integer case

2( α +1) R( ) 2 α . (2.1.8)
Since → L( ) is an even function, only even powers of are non-singular (see Remark 2.1.4 item 1). Hence, R( ) is necessarily singular. A quantitative and explicit control on this term is given, but it falls short to prove the expected 2α behaviour of (2.1.3).

Assumptions and Main Result

We will work under the following assumptions, supposed to be satisfied in the whole paper.

Assumptions 2.1.5. The random variable Z is positive, non-deterministic, and 1. If α = +∞ ( i.e., if Z 1 a.s.), then for every K 0,

(a) E[log Z] < 0 (can be -∞); (b) There exists δ > 0 such that E[Z δ ] < +∞. Introduce A = {γ ∈ [0, +∞] such that E[Z γ ] < 1}, ( 2
L( ) = K k=1 (-1) k+1 k 2k + O( 2(K+1) ). (2.1.11) 2. If α ∈ {1, 2, . . .} and if E[Z α ] = 1, then L( ) = α-1 k=1 (-1) k+1 k 2k + (-1) α+1 R( ), (2.1.12)
where R( ) is nonnegative and

2α R( ) C 2α log(1/ ), (2.1.13)
for some C > 0. The lower bound can be improved if, in addition, Z has a bounded support, to obtain, for some C c > 0, the sharper estimate c R(

) 2α log(1/ ) C. (2.1.14) 3. If α ∈ (0, +∞)\{1, 2, . . .} and if there exists γ > α such that E[Z γ ] is finite, then L( ) = α k=1 (-1) k+1 k 2k + (-1) α +1 R( ), (2.1.15)
where R( ) is nonnegative and

2 α R( ) C 2α , (2.1.16)
for some C > 0. The lower bound can be improved if, in addition, Z has a bounded support: in that case, there exists θ ∈ (α, α ) and c > 0 such that R( )

c 2θ . Remark 2.1.7. 1. The constant θ is explicit: θ = α -log E[Z α ] log Z ∞ .
2. When α ∈ (0, 1), then α = 0, so (2.1.15), and (2.1.16) should be read 2 L( ) C 2α , which is rough, and of course strongly weaker than the behaviour (2.1.7) obtained by [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF].

3. When α is finite, the lower bounds of the error in (2.1.13) and (2.1.16) assert in particular that the regular expansions (2.1.12) and (2.1.15) cannot be continued beyond K = α -1: L( ) is singular.

4. When α is not an integer, the assumption "there exists γ > α such that E[Z γ ] is finite" can be replaced by the weaker assumption "E[Z α log + Z] < +∞" (see Remark 2.5.4).

5. Suppose that α is finite and E[Z α ] < 1 (and E[Z γ ] = +∞ for every γ > α). Whether α is an integer or not, under some technical assumptions on the distribution of Z, the Lyapunov exponent is slightly regularized (see Remark 2.5.4 for a sketch of proof):

L( ) = α k=1 (-1) k+1 k 2k + (-1) α +1 R( ), 2( α +1) R( ) 2α . (2.1.17)

Strategy of the proof and structure of the paper

A classical result in the theory of product of random matrices (see e.g. [START_REF] Bougerol | Products of random matrices with application to Schrödinger operators[END_REF][START_REF] Furstenberg | Non-commuting Random Products[END_REF]) provides a semi-explicit formula for the Lyapunov exponent, involving an invariant measure for the action of the random matrices on the corresponding projective space. Here, this formula will be obtained in a direct way, and takes the form

L( ) = E[log(1 + 2 X )],
(2. 1.18) where the law of X is an invariant measure for the random transformation, on [0, +∞),

x → Z 1+x 1+ 2 x . In other words it satisfies

X (d) = Z 1 + X 1 + 2 X , (2.1.19)
where Z is independent of X (on the right hand side). Existence and uniqueness of such a random variable X will be justified in Section 2.2, as well as formula (2.1.18). A very useful uniform stochastic dominance of the random variables (X ) >0 will also be proved.

From that point on, the work will only be based on formula (2.1.18) for the Lyapunov exponent and the fixed point equation (2.1.19). Thanks to the former, the problem will readily boil down to studying X 's moments. That study can be split into two subproblems. We will know since Section 2.2 which ones of X 's moments are bounded as goes to 0 and which diverge. The two subproblems then are:

-Deriving a regular expansion for X 's bounded moments, involving an error in terms of a divergent moment of X (Sections 2.3 and 2.4);

-Estimating the divergence speed of X 's unbounded moments (Section 2.5).

The former point is addressed in Section 2.3. The analysis is based on a bootstrap procedure, based on recursive uses of the fixed point equation (2.1.19). It gives more and more precise expansions of these moments. Eventually, it will provide the regular expansion (2.1.12) or (2.1.15) with an upper bound on the error R( ), involving a divergent moment of X . That work will be generalized in the appendix 2.A, for matrices of size d, with more general entries.

That same strategy, using a bootstrap procedure to obtain a more and more precise estimate of X 's moments, can also provide a lower bound on the error, involving a divergent truncated moment of X : Section 2.4 will be devoted to that analysis.

At the end of these sections, the following theorem will be proved, which, unlike Theorem 2.1.6, does not require any extra assumption on Z (apart from Assumptions 2.1.5).

Theorem 2.1.8. Fix B > 0, and an integer K ∈ A ∪ {0}. One has, for all > 0,

L( ) = K k=1 (-1) k+1 k 2k + (-1) K+2 R K ( ), (2.1.20)
where, for all β ∈ (K, K + 1], and for some positive constants c and C β ,

c 2(K+1) E[X K+1 1 { 2 X B} ] R K ( ) C β 2β E[X β ]. (2.1.21)
Remark 2.1.9. The coefficients ( k ) are the same as in Theorem 2.1.6. The neat thing about that theorem is that, unlike the lower bound (2.1.16) of Theorem 2.1.6, the estimate (2.1.21) should be "sharp" in the following sense. If one proves that, as goes to 0, P(X c -2 ) C 2α for some positive constants c and C (the precise analysis of M 's invariant measure conducted in [START_REF] Genovese | Singular behavior of the leading Lyapunov exponent of a product of random 2 × 2 matrices[END_REF] provides such an estimate when α ∈ (0, 1)) then (2.1.21) becomes c 2α R K ( ) C 2α (with a log correction if α is an integer). It is the good order of predicted by Conjecture 2.1.2. Without such an estimate, (2.1.21) is not satisfactory yet for it is not explicit enough.

To obtain the explicit bounds given in Theorem 2.1.6, a study of the divergence speed of X 's divergent moments is needed. It is conducted in Section 2.5. The derivation of upper bounds is based a stochastic dominance found in Section 2.2 (namely X X 0 ), and on renewal theory results describing the limiting behaviour of the tail of X 0 . The lower bounds are only derived when Z is bounded. The analysis is again based on a recursive use of the fixed point equation (2.1.19). It is the point where the sharpness of the lower bound (2.1.21) of the singularity is lost. Theorem 2.1.6 is proved at the end of Section 2.5.

Existence and First Properties of the invariant measure X

In this section we prove the existence of the random variables X and derive formula (2.1.18). A first result on X 's moments is also proved: it spells out which moments of X are bounded as goes to 0 and which diverge.

We start by introducing an invariant measure of the random matrix M 0 ( = 0). It will play a central role to define the random variables X and control their moments. First I need to fix a notation for the stochastic dominance. Definition 2.2.1. The stochastic dominance will be denoted by . Formally, if X and Y are two real-valued random variables, X Y means that P(X x) P(Y x) for every x ∈ R. Equivalently, there exist two copies X and Ỹ , of X and Y respectively, such that X Ỹ almost surely.

Lemma 2.2.2. Fix a sequence (Z n ) of iid copies of Z. The series

X 0 = +∞ n=1 Z 1 • • • Z n (2.2.1)
converges almost surely. It is the unique random variable (in distribution) satisfying

X 0 (d) = Z(1 + X 0 ), (2.2.2)
with Z independent of X 0 . Moreover E[log + X 0 ] is finite; and for every γ > 0,

E[X γ 0 ] < +∞ if and only if E[Z γ ] < 1. (2.2.3) Proof. Recall that E[log Z] < 0.
The almost sure convergence of the series follows from the law of large numbers, whereby

Z 1 • • • Z n = e nE[log Z]+o(n) as n → +∞. (2.2.4)
Of course

X 0 = +∞ n=1 Z 1 • • • Z n = Z 1 1 + +∞ n=2 Z 2 • • • Z n (2.2.5)
satisfies the identity (2.2.2). Let's turn to the uniqueness. If X0 is another random variable satisfying (2.2.2), then, applying this identity N times we get

X0 (d) = N n=1 Z 1 • • • Z n + Z 1 • • • Z N X0 , (2.2.6)
where Z 1 , . . . , Z N are iid copies of Z, independent of X0 . With (2.2.4), the last term vanishes (in distribution) as N goes to +∞, whereas the first sum converges monotonically towards X 0 . So eventually, X0

= X 0 . The uniqueness is proved. Now fix γ > 0 such that E[Z γ ] < 1. We want to prove that E[X γ 0 ] is finite. If γ 1 we use Minkovsky's inequality:

E[X γ 0 ] 1/γ +∞ n=1 E[(Z 1 • • • Z n ) γ ] 1/γ = +∞ n=0 E[Z γ ] n/γ . (2.2.7)
Thus E[X γ 0 ] is finite. On the other hand, if γ ∈ (0, 1), then for all x, y 0, (x + y)

γ x γ + y γ . So E[X γ 0 ] +∞ n=0 E[Z γ ] n , (2.2.8) which is again finite. Now, if E[Z γ ] 1, then with the identity (2.2.2), E[X γ 0 ] = E[Z γ ]E[(1 + X 0 ) γ ] E[(1 + X 0 ) γ ],
(2.2.9)

which can hold only if E[X γ 0 ] = +∞ (or γ = 0). Eventually, pick γ ∈ A so that E[Z γ ] < 1.
With the foregoing, we then know that E[X γ 0 ] < +∞. Thus, by Jensen's inequality, E[log + X 0 ] is finite.

The next lemma provides the existence of the random variables X and the desired formula for the Lyapunov exponent.

Lemma 2.2.3. For all > 0, there exists a non-negative random variable X , unique in distribution, such that

X (d) = Z 1 + X 1 + 2 X , (2.2.10)
with Z independent of X . Moreover, for every ∈ (0, 1], Z X X 0 . The Lyapunov exponent can be written

L( ) = E[log(1 + 2 X )]. (2.2.11)
Furthermore, for every x, y ∈ R 2 with nonnegative entries,

1 n log x, M n, • • • M 1, y -→ n→+∞ L( ) a.s. and in L 1 .
(2.2.12)

Remark 2.2.4. There could be other distributions, supported on R, satisfying (2.2.10). We only claim uniqueness for non-negative invariant measure. However, if Z does not have a finite support, then one can prove, using classical results of products of random matrices (see [START_REF] Furstenberg | Non-commuting Random Products[END_REF] or [START_REF] Bougerol | Products of random matrices with application to Schrödinger operators[END_REF]Chapter 3]), that there exists a unique invariant measure on R. With Lemma 2.2.3, we know that it must be supported on R + .

In what follows, X will always denote the unique non-negative invariant random variable of Lemma 2.2.3.

Proof. We begin with the proof of the existence, for which we use a standard procedure. Fix an iid sequence (Z n ) of copies of Z, set x 0 = 0 and define recursively the random variables

x n+1 = Z n+1 1 + x n 1 + 2 x n . ( 2

.2.13)

Denote by ν n the distribution of the random variable x n and consider the measure

ρ N = 1 N N -1 n=0 ν n .
Observe that for any n 0, x n is nonnegative and x n+1 Z n+1 (1 + x n ). Thus, by an easy induction,

0 x n n-1 k=0 Z n • • • Z n-k X 0 : (2.2.14)
the random variables x n are uniformly bounded by X 0 . Consequently the sequence (ρ N ) is tight. Pick a limit point ρ ∞ of that sequence and fix a random variable X with distribution ρ ∞ . The limit distribution ρ ∞ must be invariant under the random transformation (2.2.13).

In other words it must satisfy (2.2.10). The existence of an invariant measure supported on R + is proved. Incidentally we obtained X X 0 . As for the stochastic lower bound X Z, it directly follows from the identity (2.2.10) when 1. To deal with the uniqueness, assume that X (0) and Y (0) are two such random variables and fix an iid sequence (Z n ) of copies of Z, independent of X (0) and Y (0) . We introduce, for n 0,

X (n+1) = Z n+1 1 + X (n) 1 + 2 X (n) , Y (n+1) = Z n+1 1 + Y (n) 1 + 2 Y (n) . (2.2.15)
Observe that, almost surely, 0) . The uniqueness follows. Then ρ N actually converges (without extraction) towards X 's distribution. We are left with the proof of formula (2.2.11). Thanks to a result by H. Hennion [START_REF]Limit theorems for products of positive random matrices[END_REF], since M 's entries are positive, the convergence (2.2.12) holds. On the other hand, for every n 0,

|X (n+1) -Y (n+1) | = Z n+1 (1 -2 )|X (n) -Y (n) | (1 + 2 X (n) )(1 + 2 Y (n) ) Z n+1 |X (n) -Y (n) |. ( 2
(d) = X (0) and Y (n) (d) = Y (n)
M n, • • • M 1, 1 X (0) = n-1 k=0 (1 + 2 X (k) ) 1 X (n) (2.2.17)
So, by taking the log and the expectation,

1 n E[log M n, • • • M 1, t (1, X (0) ) ] = E[log(1 + 2 X )] + 1 n E[log (1, X ) ]. (2.2.18) Since E[log + X ] E[log + X 0 ] is finite (Lemma 2.2.
2), the last term vanishes as n goes to +∞. On the other hand, one has, for every n 0,

(1, 0)M n, • • • M 1, t (1, 0) M n, • • • M 1, t (1, X (0) ) M n, • • • M 1, (1 + X (0) ). (2.2.19)
Since we know that both the lower and upper bounds goes to L( ) (after taking log and expectation) as n goes to +∞, almost surely and in L 1 , we get the result.

Remark 2.2.5. Here the most usual result [START_REF] Furstenberg | Non-commuting Random Products[END_REF] does not apply (no strong irreducibility) and we invoked [START_REF]Limit theorems for products of positive random matrices[END_REF]. Actually, formula (2.2.11) can also be proved using a classical result by H. Furstenberg and Y. Kifer [47, Corollary of Theorem 3.10], which gives an explicit formula for the Lyapunov exponent in terms of invariant measures as soon as M is an invertible random matrix of size d×d with no deterministic proper invariant subspace. We could also have used the convergence ρ n → law of X , to prove (2.2.11) and (2.2.12) without using H. Hennion's results.

Remark 2.2.6. If one notes that the map → 1+x 1+ 2 x . is monotone, one obtains, with the previous construction, that the random variables X are stochastically decreasing with : for all > 0 one has X X X 0 .

Lemma 2.2.7. X → X 0 in distribution when → 0.

Proof. The stochastic dominance X X 0 ensures that the family of random variables (X ) >0 is tight. Consider a limit point X0 of X as goes to 0. Since X satisfies the identity (2.2.10), the limit point X0 must satisfy X0 (d) = Z(1 + X0 ). That means, using Lemma 2.2.2, that X 0 is the only possible limit point of X as goes to 0. The convergence of X towards X 0 (in distribution) follows.

Using classical integration theorems, one readily obtains the following limiting behaviour of X 's moments, or truncated moments, which will be needed in the proof of Theorem 2.1.6.

Corollary 2.2.8. For any γ > 0,

1. If E[Z γ ] < 1 then, as goes to 0, E[X γ ] = O(1). 2. If E[Z γ ] 1 then for any B > 0, E X γ 1 { 2 X B} -→ →0 +∞.
(2.2.20)

Proof. Recall that E[X γ 0 ] is finite if and only if E[Z γ ] < 1 (Lemma 2.2.2). With the stochastic dominance X X 0 provided by Lemma 2.2.3, we get E[X γ ] = O(1) when E[Z γ ] < 1. On the other hand, if E[Z γ ] 1, then E[X γ 0 ] = +∞ (Lemma 2.2.
2). Thanks to Skorokhod's representation theorem and the convergence in distribution provided by Lemma 2.2.7, there exists representatives X and X0 such that X → X0 almost surely. And then, with Fatou's lemma, for any B > 0.

lim inf →0 E Xγ 1 { 2 X B} E lim inf →0 Xγ 1 { 2 X B} = E [X γ 0 ] = +∞. (2.2.21)

Regular Expansion (Theorem 2.1.8: upper bound)

In this section we prove the existence of a regular expansion for the Lyapunov exponent L( ). We also lay out the method, which will be used twice more: for the generalization of this result in Appendix 2.A and in Section 2.4 to obtain the lower bound of the error. It is based on the study of a regular expansion for the moments of X which are bounded as goes to 0. Let us first state the main result of the section.

L( ) = K k=1 (-1) k+1 k 2k + O( 2β E[X β ]), (2.3.1)
where, for k K, k is a positive rational function of

E[Z], . . . , E[Z k ]. Remark 2.3.2. With some extra effort, the domination O( 2β E[X β ]) can be replaced by O(E[( 2 X ) β 1 ∧ ( 2 X ) β 2 ]) for any β 1 , β 1 ∈ [K, K + 1].
To achieve this, replace the upper bounds x β (resp. x δ ) by

x β 1 ∧ x β 2 (resp. x δ 1 ∧ x δ 2
). It will only be needed to explain (see Remark 2.5.4) some generalizations discussed in Remark 2.1.7.

Proof. We use identity L(

) = E[log(1 + 2 X )] (Lemma 2.2.
3) and expand the logarithm. There exists C > 0 such that for all x 0,

log(1 + x) - K j=1 (-1) j+1 j x j Cx β . (2.3.2) Consequently L( ) = K j=1 (-1) j+1 j 2j E[X j ] + O 2β E[X β ] . (2.3.3)
Lemma 2.3.3. For all l K, the following expansion holds,

E[X l ] = K-l k=0 (-1) k g l,k 2k + O( 2(β-l) E[X β ]), (2.3.4)
where, for all l 1 and for all k 0, the coefficient g l,k is a positive rational function of the moments

E[Z], . . . , E[Z l+k ].
We first admit Lemma 2.3.3 and conclude the proof of Proposition 2.3.1. The substitution of (2.3.4) into (2.3.3) yields

L( ) = K j=1 K-j k=0 (-1) j+k+1 j 2(j+k) g j,k + O 2β E[X β ] . (2.3.5)
It can be rewritten

L( ) = K s=1 (-1) s+1 s 2s + O 2β E[X β ] , (2.3.6)
with

s = K j=1 K-j k=0 g j,k j 1 {j+k=s} , (2.3.7)
and s is a positive rational function of E[Z], . . . , E[Z s ] by inspection.

We are left with the proof of Lemma 2.3.3, for which we briefly explain the strategy. Write the identity

E[X k ] = E[Z k ]E 1 + X 1 + 2 X k . (2.3.8)
Then by expanding the denominator one gets

E[X k ] = E[Z k ] n j=0 -k j 2j E (1 + X ) k X j + Remainder. (2.3.9)
It gives a relation between the moments of X which will be used via a bootstrap procedure: the substitution of a regular expansion for X 's first moments into (2.3.9) will provide a more precise expansion of E[X k ]. That new expansion will in turn be injected into (2.3.9) (for another k), to obtain a more precise regular expansion for that other moment, et cetera. Of course that procedure should be done in a specific order. Doing it rigorously will require a double induction, on k and the length of the expansions. Let's now proceed to the detailed proof.

Proof of Lemma 2.3.3. Set δ = β -K. We prove, using a course-of-values double induction with the lexicographic order on (m, j), that if j + m K, then E[X j ] has an expansion up to the order 2m :

E[X j ] = m k=0 (-1) k g j,k 2k + O 2(m+δ) E[X β ] , (2.3.10)
where for every j 1 and k 0, the coefficient g j,k is a positive rational function of

E[Z], . . . , E[Z j+k ].
Of course E[X 0 ] admits such an expansion, up to any order. All that remains is the inductive step. Fix l 1 and n 0 such that l + n K and suppose that (2.3.10) holds (A) for all j K and m (n -1) ∧ (Kj);

(B) for all j l -1, and m n.

We want to show that it also holds for (j, m) = (l, n). To this end, write

E[X l ] = E Z 1 + X 1 + 2 X l = E[Z l ] l r=0 l r E X r (1 + 2 X ) l .
(2.3.11)

We want to expand the denominator with respect to . Let C > 0 be such that for any x 0 and l, m K,

1 (1 + x) l - m i=0 -l i x i Cx m+δ . (2.3.12)
Thus, for every r l,

E X r (1 + 2 X ) l - n i=0 -l i 2i E[X i+r ] C 2(n+δ) E[X r+n+δ ] C 2(n+δ) max 0 k K E[X k+δ ].
(2.3.13)

Actually max k K E[X k+δ ] = O(E[X β ]). (2.3.14) Indeed, if 1 k K -1, then E[Z k+δ ] < 1, so E[X k+δ ] E[X k+δ 0 ] < +∞ (Lemmas 2.2.

and 2.2.3). On the other hand E[X

K+δ ] = E[X β ] E[Z β ] > 0 (Lemma 2.2.
3). Thus, with (2.3.13) and (2.3.14), we can write, for every r l,

E X r (1 + 2 X ) l = n i=0 -l i 2i E[X i+r ] + O( 2(n+δ) E[X β ]). (2.3.15)
And then, injecting it into (2.3.11), we get

E[X l ] = E[Z l ] l r=0 l r n i=0 -l i 2i E X i+r + O( 2(n+δ) E[X β ]). (2.3.16)
We then isolate the term "(i, r) = (0, l)" -that is E[Z l ]E[X l ] -on the left-hand side and divide by

1 -E[Z l ], to get E[X l ] = E[Z l ] 1 -E[Z l ] 0 r l, 0 i n (i,r) =(0,l) l r -l i 2i E X i+r + O( 2(n+δ) E[X β ]).
(2.3.17)

We claim that the induction hypothesis provides expansions for all these terms, up to the required order. The induction hypothesis (2.3.10) on E[X i+r ] (induction hypothesis with j = i + r and m = ni, which is contained in the item (B) if i = 0 and in the item (A) if i 1), states that

E[X i+r ] = n-i k=0 2k (-1) k g i+r,k + O 2(n-i+δ) E[X β ] . (2.3.18)
We then inject it into (2.3.17). It yields

E[X l ] = E[Z l ] 1 -E[Z l ] 0 r l, 0 i n (i,r) =(0,l) l r -l i 2i n-i k=0 2k (-1) k g i+r,k + O 2(n+δ) E[X β ] . (2.3.19)
One can already observe that it is a regular expansion of E[X l ] up to the order n, as expected. The following lines intend to derive a recursive formula for g l,k so as to check its sign. First note that -l i = (-1) i l + i -1 i .

(2.3.20)

Thus (2.3.19) becomes E[X l ] = E[Z l ] 1 -E[Z l ] 0 r l, 0 i n (i,r) =(0,l) 0 k n-i l r l + i -1 i 2(k+i) (-1) k+i g i+r,k + O 2(n+δ) E[X β ] . (2.3.21)
Eventually, it can be written as

E[X l ] = n s=0 (-1) s g l,s 2s + O 2(n+δ) E[X β ] , (2.3.22)
with, for every s n,

g l,s = E[Z l ] 1 -E[Z l ] 0 r l, 0 i n (i,r) =(0,l) 0 k n-i l r l + i -1 i g i+r,k 1 {i+k=s} . (2.3.23)
Thanks to the induction hypothesis, it is a positive rational function of

E[Z], ..., E[Z l+n ].
The inductive step is proved, and the lemma follows.

Theorem 2.1.8: lower bound on the error

We prove here the lower bound on the error given in Theorem 2.1.8, formula (2.1.21). We already saw in Proposition 2.3.1's proof, when we studied the signs in front of the coefficients k or g l,k , that when expanding the algebraic fractions (1 + 2 X ) -r , the term 2n always comes with the sign (-1) n . The same occurs for the error, at each step, at the order K+1 : it comes with the sign (-1) K+1 . As a result, the error terms, which invariably accumulate with the same sign, effectively add up and cannot offset one another. In practice, these error terms can also be bounded from below. It yields the next result.

Proposition 2.4.1. Fix an integer K ∈ A ∪ {0} and B > 0. There exists c > 0 such that, for all > 0,

(-1) K+2 L( ) - K k=1 (-1) k+1 k 2k c 2(K+1) E[X K+1 1 { 2 X B} ], (2.4.1)
where the coefficients ( k ) are the same as in Proposition 2.3.1.

Unsurprisingly, a similar scheme as in Proposition 2.3.1's proof will be used. We will proceed to a double induction, corresponding to an underlying bootstrap procedure. The only actual difference compared to Section 2.3 is that the estimate (2.3.12) is replaced by the lower bound

1 (1 + x) m - r i=0 -m i x i C(-x) r+1 1 {x B} . (2.4.2)
We begin with the equivalent of Lemma 2.3.3 in this new perspective.

Lemma 2.4.2. Fix an integer K ∈ A ∪ {0} and B > 0. There exists c > 0 such that for all 1 l K, and 0 n Kl, the following holds, for the same real coefficients (g l,k ) as in Lemma 2.3.3:

E[X l ] - n k=0 (-1) k g l,k 2k c(-1) n+1 2(n+1) E[X l+n 1 { 2 X B} ] (2.4.3)
if n + 1 is even, and the same with an inequality in the opposite direction if n + 1 is odd.

Proof. If K = 0 the statement is empty, so suppose K 1. It will be useful to recall formula (2.3.20). Fix B > 0. There exists C > 0 such that for all 1 l K + 1 and n K + 1, and for all x 0,

1 (1 + x) l - n-1 i=0 l + i -1 i (-x) i C(-x) n 1 {x B} if n is even, C(-x) n 1 {x B} if n is odd. (2.4.4)
As in Lemma 2.3.3, we carry out a proof by course-of-values double induction. More precisely, set

C := C min 1 l K E[Z l ] 1 -E[Z l ] . (2.4.5)
We prove that if j 1, m 0 and j + m K + 1 then

E[X j ] - m-1 k=0 2k (-1) k g j,k C(-1) m 2m E[X j+m 1 { 2 X B} ] (2.4.6)
if m is even (and of course (-1) m = 1); and the same with an inequality in the opposite direction if m is odd (then (-1) m = -1). The base case m = 0 is immediate. For the inductive step, we fix l 1, n 1 such that l+n K +1 and we suppose that (2.4.6) holds for all (j, m) with m n -1 and 1 j K + 1m, and for all (j, n) with 1 j l -1.

We want to prove (2.4.6) for (j, m) = (l, n). For the sake of simplicity, the proof will only be written for n even (inequalities would be in the opposite direction if n is odd). First write the identity

E[X l ] = E Z 1 + X 1 + 2 X l = E[Z l ] l r=0 l r E X r (1 + 2 X ) l .
(2.4.7)

Using (2.4.4) we get,

E[X l ] E[Z l ] l r=0 l r n-1 i=0 l + i -1 i (-1) i 2i E[X i+r ] C(-1) n 2n E[X n+r 1 { 2 X B} ] .
(2.4.8)

We subtract the term

E[Z l ]E[X l ] (term (i, r) = (0, l)) and divide by 1 -E[Z l ] (which is positive) to obtain E[X l ] E[Z l ] 1 -E[Z l ] 0 r l, 0 i n-1 (i,r) =(0,l) l r l + i -1 i (-1) i 2i E[X i+r ] + C(-1) n 2n E[X n+r 1 { 2 X B} ] .
(2.4.9)

We use the induction hypothesis on E[X i+r ] (induction hypothesis (2.4.6) with j = i + r and m = ni), that is

E[X i+r ] - n-i-1 k=0 2k (-1) k g i+r,k C(-1) n-i 2(n-i) E[X r+n 1 { 2 X B} ], (2.4.10)
if ni is even, and the opposite if it is odd. In any case, injecting these lower bounds into (2.4.9) yields

E[X l ] E[Z l ] 1 -E[Z l ] 0 r l, 0 i n-1 (i,r) =(0,l) l r l + i -1 i × × (-1) i 2i n-i-1 k=0 2k (-1) k g i+r,k + C(-1) n 2n E[X n+r 1 { 2 X B} ] + C(-1) n 2n E[X n+r 1 { 2 X B} ] .
(2.4.11)

The first terms corresponds to the regular part already found in Lemma 2.3.3 equations (2.3.22) and (2.3.23); the second line contains the 2n -terms which we want to bound from below:

E[X l ] n-1 s=0 2s (-1) k g l,s + (-1) n 2n Q n , (2.4.12)
with

Q n = E[Z l ] 1 -E[Z l ] 0 r l, 0 i n-1 (i,r) =(0,l) l r l + i -1 i C + C E[X n+r 1 { 2 X B} ]. (2.4.13)
Since all the terms in Q n are non-negative, it is larger than any of them

Q n E[Z l ] 1 -E[Z l ] CE X n+l 1 { 2 X B} CE X n+l 1 { 2 X B} . (2.4.14)
This concludes the proof of the inductive step and thus the proof of the lemma.

Proof of Proposition 2.4.1. Let c = c (B, K) > 0 be such that for all x 0, log(1 + x)

K l=1 (-1) l+1 x l l + c (-1) K+2 x K+1 1 {x B} (2.4.15)
if K is even; and the same with an inequality in the opposite direction if K is odd.

For the sake of simplicity we suppose that K is even in what follows. Writing L( ) = E log(1 + 2 X ), we get

L( ) K l=1 (-1) l+1 l 2l E[X l ] + c (-1) K+2 2(K+1) E[X K+1 1 { 2 X B} ], (2.4.16)
and Lemma 2.4.2 provides a lower bound for each term in the sum: for every 1 l K,

(-1) l+1 2l E[X l ] (-1) l+1 2l K-l k=0 2k (-1) k g l,k + c(-1) l+1 (-1) K+1-l 2(K+1) E[X K+1 1 { 2 X B} ].
(2.4.17)

The conclusion results from the latter two inequalities.

Limiting Behaviour of X 's divergent moments

First note that Theorem 2.1.8 is an immediate consequence of Propositions 2.3.1 and 2.4.1. The goal of this section is to obtain estimates of the error R K ( ), for which we now have

c 2(K+1) E[X K+1 1 { 2 X B} ] R K ( ) C β 2β E[X β ]. (2.5.1)
In order to give explicit estimates of R K ( ) in terms of powers of , one needs to understand the limiting behaviour of X 's moments (or truncated moments). The issue was partially addressed by Corollary 2.2.8, which pinpointed the regimes of convergence or divergence of these moments. Namely E[X γ ] is bounded as goes to

0 if E[Z γ ] < 1 and diverges if E[Z γ ] 1.
In the following section we address the issue of the divergence speed when

E[Z γ ] 1.
The first paragraph, based on renewal theory results, describing the heavy tail of X 0 , will provide upper bounds for X 's divergent moments. The second paragraph will give lower bounds for these moments under the restriction that Z is bounded.

Upper Bounds

We will need the following result, which combine results by H. Kesten and A. K. Grincevičius depending if log Z has an arithmetic support or not (see [START_REF]Implicit renewal theory in the arithmetic case[END_REF]Theorems 1,[START_REF]Path properties of the disordered pinning model in the delocalized regime[END_REF] for a review).

Lemma 2.5.1. If E[Z α log + Z] < +∞, then, as x goes to +∞,

P(X 0 x) = O(x -α ).
(2.5.

2)

It readily gives the next two results. They provide explicit upper bounds for the speed of divergence of X 's moments. If you believe Conjecture 2.1.2, these upper bounds (except the first one when E[Z α ] < 1) are of the good order of . The first one will be used for α ∈ {1, 2, . . .} whereas the second will be needed when α is not an integer.

Lemma 2.5.2. If E[Z α log + Z] < +∞, then, as goes to 0, E[X α ] = O (log(1/ )) . (2.5.3)
Proof. The identity

X (d) = Z 1+X 1+ 2 X yields, for γ 0, E[X γ ] = E[Z γ ]E 1 + X 1 + 2 X γ E[Z γ ]E (1 + X ) ∧ -2 γ E[Z γ ]E (1 + X 0 ) ∧ -2 γ .
(2.5.4)

It can be rewritten

E[X γ ] E[Z γ ] γ -2 0 x γ-1 P(X 0 > x -1)dx + -2γ P(X 0 -2 -1) . (2.5.5)
With γ = α, Lemma 2.5.1 gives upper bounds for these two terms:

-2α P(X 0 -2 -1) = O(1) (2.5.6)
and

-2 0 x α-1 P(X 0 > x -1)dx = O(log(1/ )).
(2.5.7)

Lemma 2.5.3. Fix γ > α and assume that E[Z γ ] is finite. Then, as goes to 0,

E[X γ ] = O( 2α-2γ
).

(2.5.8)

Proof. We reuse inequality (2.5.5). Lemma 2.5.1, which applies here, yields

-2γ P(X 0 -2 -1) = O( 2α-2γ ) (2.5.9)
and -For the item 4, note that when E[Z α log + Z] < +∞, Lemma 2.5.1, and the techniques used for Lemmas 2.5.2 and 2.5.

-2 0 x γ-1 P(X 0 > x -1)dx = O( 2α-2γ
3, yield E[( 2 X ) α ∧ ( 2 X ) α ] = O( 2α ).
-For the item 5, an alternative version of Lemma 2.5.1 should be used: when

E[Z α ] < 1,
and under some extra technical assumptions, the estimate P(X 0 x) = o(x -α ) holds (see [START_REF] Kevei | A note on the Kesten-Grincevičius-Goldie theorem[END_REF]Theorem 1.3] and [START_REF]Implicit renewal theory in the arithmetic case[END_REF]Theorem 8]). We deduce

E[( 2 X ) α ∧( 2 X ) α ] = o( 2α ).

Lower Bounds when Z is bounded

We start with a quite general, albeit quite complex, lower bound for X 's moments.

E[X γ ] N k=1 E Z γ 1 {Z B} k exp -τ 2 B k P(X C).
(2.5.11)

Proof. Let X (N ) be a copy of X and (Z k ) be iid copies of Z, independent of X (N ) . Define recursively, for 0 k N -1,

X (k) = Z k+1 1 + X (k+1) 1 + 2 X (k+1) .
(2.5.12)

For every k N , one has

X (k) (d)
= X . On the other hand, one can derive the following lower bounds:

(X (0) ) γ = Z γ 1 (1 + X (1) ) γ (1 + 2 X (1) ) γ Z γ 1 (1 + 2 X (1) ) γ + Z γ 1 (1 + 2 X (1) ) γ (X (1) ) γ .
(2.5.13)

Here the condition γ 1 is used through the convexity inequality (1 + x) γ 1 + x γ . Then, inductively,

(X (0) ) γ Z γ 1 (1 + 2 X (1) ) γ + Z γ 1 (1 + 2 X (1) ) γ Z γ 2 (1 + 2 X (2) ) γ + • • • + N j=1 Z γ j (1 + 2 X (j) ) γ .
(2.5.14)

By taking the expectation we get

E[X γ ] N k=1 E   k j=1 Z γ j (1 + 2 X (j) ) γ   . (2.5.15) If X (k)
C and Z k B, then, with definition (2.5.12), X

(k-1) B(1+C). So, inductively, if X (k)
C, and Z 0 , . . . , Z k B, then, for every j k,

X (k-j) j i=1 B i + B j C B j B B -1 + C = B j σ, (2.5.16
)

with σ = B B-1 + C. Thus, k j=1 Z γ j (1 + 2 X (j) ) γ   k j=1 Z γ j 1 {Z j B} (1 + σ 2 B k-j ) γ   1 {X (k) C} .
(2.5.17)

We compute

k j=1 1 (1 + σ 2 B k-j ) γ exp   -σγ 2 k j=1 B k-j   exp - σγ B -1 2 B k = exp -τ 2 B k .
(2.5.18)

Taking the expectation in (2.5.17) and using that X (k) and Z 1 , . . . , Z k are independent, we obtain

E   k j=1 Z γ j (1 + 2 X (j) ) γ   E Z γ 1 {Z B} k exp -τ 2 B k P(X C). (2.5.19)
The conclusion follows by injecting this lower bound into (2.5.15).

One could expect to use this general lower bound for any given Z. However it only gives satisfactory results when Z is bounded. In that case we can get rid of the indicator 1 {Z≤B} in (2.5.11). Lemma 2.5.6. If Z has a bounded support then X -2 Z L ∞ almost surely.

Proof. It is an immediate consequence of the invariance identity

X (d) = Z 1+X 1+ 2 X and of the inequality 1+x 1+ 2 x
-2 , which holds for every x 0. Lemma 2.5.6 justifies that we only study X 's moments instead of its truncated moments: as long as B is chosen larger than Z L ∞ one has

E X K+1 1 { 2 X B} = E X K+1 .
(2.5.20)

In the next two lemmas we give a lower bound for X 's moments when Z is bounded.

In that instance, note that E[Z α ] = 1: the set A cannot takes the form A = (0, α]. Lemma 2.5.7 will be used if α is an integer, and Lemma 2.5.9 when α is not an integer. However, both of them hold true regardless of the nature of α.

Lemma 2.5.7. For α 1, if Z has a bounded support then, for some c > 0, and sufficiently small, E[X α ] c log(1/ ).

(2.5.21)

Proof. Recall that since Z is bounded, E[Z α ] = 1. Choose γ = α and B = Z ∞ in Lemma 2.5.5 to get E[X α ] N k=1 E [Z α ] k exp -τ 2 B k P(X C) N exp -τ 2 B N P(X C).
(2.5.22)

First note that, thanks to Lemma 2.2.7, Lemma 2.5.9. If Z is bounded, and if γ 1 is such that E[Z γ ] > 1, then, for some c > 0, and for sufficiently small,

P(X C) -→ P(X 0 C), (2.5.23) which is positive if C is large enough. Choosing N = N = 2 1 log B log 1 , we obtain E[X α ] N exp(-τ )P(X C) c log(1/ ). ( 2
E[X γ ] c -2η , where η = log E[Z γ ] log Z ∞ ∈ (0, γ -α).
(2.5.25)

Proof. Set B = Z L ∞ in Lemma 2.5.5 to get E[X γ ] N k=1 E [Z γ ] k exp -τ 2 B k P(X C) E [Z γ ] N exp -τ 2 B N P(X C).
(2.5.26) Proof. We recall here the upper and lower bounds provided by Theorem 2.1.8:

Choosing again N = 2 1 log B log 1 , we obtain E[X γ ] E [Z γ ] N exp (-τ ) P(X C) c -2η . ( 2
c 2(K+1) E[X K+1 1 { 2 X B} ] R K ( ) C β 2β E[X β ], (2.5.28)
If α = +∞, then, thanks to Lemma 2.2.3

R K ( ) C K+1 2(K+1) E[X K+1 ] 2(K+1) E[X K+1 0 ]. (2.5.29) Since E[X K+1 0 ] is finite (Lemma 2.2.
2), the result (2.1.11) follows. From now on we suppose that α is finite and E[Z α ] = 1 and we set

K = α -1. By Corollary 2.2.8, R( ) c 2(K+1) E[X K+1 1 { 2 X B} ] 2(K+1) .
(2.5.30)

If α is an integer then the lower and upper bounds given by (2.1.13) or (2.1.14) follow from Lemmas 2.5.2 and 2.5.7. If α is not an integer then the lower and upper bounds (2.1.16) given by Theorem 2.1.6 are a consequence of Lemmas 2.5.3 (with γ such that E[Z γ ] < +∞) and 2.5.9 (with γ = K + 1).

2.A Appendix: Generalization to higher dimension

The techniques developed in the previous sections are sufficiently robust to be used in more general settings. We apply them to a square matrix of size d + 1 which is a perturbation of a matrix alike Diag(1, Z), which still have a preferred direction. Since the proofs are only slightly different from the previous sections, they will be only sketched in this appendix. We will just point out the arguments that must be adapted and many details will be omitted.

We now consider the (d + 1) × (d + 1) matrix

M = 1 L C N , (2.A.1)
where L and C are random vectors of size d, and N is a random matrix, of size d × d.

We are still interested in the Lyapunov exponent, defined by the limit

L( ) = lim n→+∞ 1 n log M n, • • • M 1, , (2.

A.2)

where (M k, ) k 1 are iid copies of M . This limit exists almost surely and is deterministic (see again [START_REF] Furstenberg | Products of random matrices[END_REF]) as soon as for every > 0, E[log + M ] < +∞. We derive in this section a regular expansion for L( ), alike the expansion provided by Proposition 2.3.1 in the previous setting. However, no lower bound on the error will be given here. We start by deriving a formula alike "L( ) = E[log(1 + 2 X )]" (Lemma 2.A.3).

In the whole section • will denote a given norm on R d or R d+1 , as well as the induced operator norm on M d (R) or M d+1 (R). On another note, if x, y ∈ R d , we will write x y if the inequality holds coordinatewise. Similarly the stochastic dominance will be extended to random vectors: X Y means that there exists a copy X of X and a copy Ỹ of Y satisfying X Ỹ almost surely (coordinatewise).

Let's introduce the assumptions under which we will work in the section. Observe that under these assumptions, the condition E[log + M ] < +∞ is fulfilled so the Lyapunov exponent is well defined. Assumptions 2.A.1. We assume that the following holds, for every ∈ (0, 0 ). Before deriving the formula for the Lyapunov exponent, we introduce the random vector Y , which will play the same role as X 0 in our new setting (except that here it will depend on ). Namely it will be used through stochastic dominances.

Lemma 2.A.2. Fix ∈ (0, 0 ) and let (N ,k , C ,k ) be iid copies of (N , C ). The series The next lemma provides the desired formula for L( ).

Y = +∞ n=0 N ,1 . . . N ,n-1 C ,n (2.
Lemma 2.A.3. There exists a random vector X ∈ R d , with non-negative entries, satisfying

1 X (d) = 1 L C N 1 X
in the projective space P d (R), (2.A.5)

or equivalently,

X (d) = C + N X 1 + 2 L X , (2.A.6)
where C , N and L are the blocks of the random matrix M , independent of X . One has X Y . Moreover the Lyapunov exponent can be written as

L( ) = E[log(1 + 2 L X )]. (2.A.7)
And for every x, y ∈ R d+1 + ,

L( ) = lim n→∞ 1 n log x, M n, • • • M 1, y . (2.A.8)
Proof. The method is the same as in Lemma 2.2.3's proof for 2 × 2 matrices. We fix iid copies (M ,n ) of M and set x 0 = 0 R d . Then define inductively, for n 0, the random variables

x n+1 = C ,n + N ,n x n 1 + 2 L ,n x n .
(2.A.9)

Observe that since all the vectors have non-negative entries, one can write, coordinatewise,

x n+1 C ,n + N ,n x n .
(2.A.10) So, by an easy induction, x n Y for every n 0. The end of the proof is the same as for Lemma 2.2.3. We do not reiterate all the details here. Just note that we do not claim the uniqueness of a non-negative solution to (2.A.6) and that Assumption 2.A.1 (a) is a sufficient condition for H. Hennion's result to apply.

To state our main result, and more precisely to formulate its premises, some multiindex notations will be required, which we set in the next lines. The norm of a multi-index λ ∈ N d will be denoted by |λ|:

|λ|

:= λ 1 + . . . λ d . (2.A.11)
For every l 0, there are l+d-1 d-1

multi-indices with norm l: it is the number of (weak) compositions of l into d non-negative integers. For a vector x ∈ R d and a multi-index λ ∈ N d , we define the multi-index power

x λ = x λ 1 1 × • • • × x λ d d .
(2.A.12)

Similarly, for a matrix

A ∈ M d (R) and a multi-index ω ∈ N d 2 M d (N), define A ω = i,j (A i,j ) ω i,j and |ω| = i,j ω i,j . (2.

A.13)

There should be no confusion with a standard matrix power since ω is a multi-index. For l 0, consider the square matrix G (l) with size l+d-1 d-1 , whose elements are

G (l) λ,λ = ω∈N d 2 j ω i,j =λ i i ω i,j =λ j lim →0 E [N ω ] , for λ, λ ∈ N d such that |λ| = |λ | = l. (2.A.14)
Note that all the multi-indices ω in the sum have norm |ω| = l. The matrix G (l) will play a similar role as E[Z l ] in this generalized context. Of course these matrices, which require the existence of lim →0 E [N ω ], are not always defined.

We have set enough notations to state the generalization of Proposition 2.3.1, giving a regular expansion of the Lyapunov exponent L( ).

Proposition 2.A.4. Fix K 0 and β ∈ (K, K + 1]. Suppose that 1. For all multi-indices λ, µ ∈ N d , ω ∈ N d 2 such that l = |λ| + |µ| + |ω| K, E[L λ C µ N ω ]
is finite and admits a regular expansion, as goes to 0, up to the order 2(Kl):

E[L λ C µ N ω ] = 2(K-l) r=0 c λ,µ,ω,r r + O( 2(β-l) ); (2.A.15)
2. For all 1 l K, the matrix I -G (l) is invertible;

3. lim sup →0 E[ L β ] is finite.
Then there exist real coefficients q 2 , . . . q 2K such that, as goes to 0,

L( ) = 2K k=2 q k k + O( 2β E[1 + X β ]). ( 2
E[N ω ] = c 0,0,ω,0 . The invertibility of I -G (l) is the counterpart of the assumption "E[Z l ] < 1" in Proposition 2.3.1.
Proof. The same proof as for Proposition 2.3.1 works: one expands the logarithm:

E[log(1 + 2 L X )] = K k=0 (-1) k+1 k 2k E[(L X ) k ] + O( 2β E[(L X ) β ]) = K k=0 (-1) k+1 k 2k 1 r 1 ,...,r k d E k i=1 (L ) r i E k i=1 (X ) r i + O( 2β E[(L X ) β ]), (2.A.17)
where x (r) stands for the r th coordinate of x. Note that ):

E[(L X ) β ] E[ L β ]E[ X β ] CE[ X β ], (2.
E[X λ ] = 2(K-l) k=0 k g λ,k + O( 2(β-l) E[1 + X β ]). (2.A.19)
Sketch of proof of Lemma 2.A.8. We can follow the same proof as for Lemma 2.3.3. We go back to that proof to understand how the present one must be adjusted. The only point which merits special attention is the line (2.3.17) where the term E[Z l ]E[X l ] is isolated on the left-hand side. That line could be summarized as follow: we wrote

E[X l ] = E[Z l ]E[X l ] + (♦ l ), (2.A.20)
where (♦ l ) stands for all the terms in the expansion of E[X l ] for which the induction hypothesis provided an expansion up to the required order. To be explicit,

(♦ l ) = E[Z l ] 0 j l, 0 i n (i,j) =(0,l) l j -l i 2i E X i+j + O( 2(n+δ) E[X β ]). (2.A.21)
Then we could conclude by writing

E[X l ] = 1 1 -E[Z l ] (♦ l ), (2.A.22)
and applying the induction hypothesis. That is where was used the condition "E[Z l ] < 1"

(actually E[Z l ] = 1 was enough), and this is where will be used the invertibility of 1 -G (l) .

In our generalized setting, we still carry out an induction on (n, l = |λ|) (equipped with the lexicographic order). For the inductive step, there are a lot of multi-indices with given norm l. They will be solved simultaneously, by writing a joint system satisfied by all these multi-indices moments E[X λ ] with |λ| = l. To this end, use the identity

E[X λ ] = E C + N X 1 + 2 L X λ = E (C + N X ) λ (1 + 2 L X ) l . (2.A.23)
Then develop the denominator

E[X λ ] = E   (C + N X ) λ   n j=0 -l j 2j (L X ) j + 2(n+δ) O((L X ) n+δ )     . (2.A.24)
Eventually, after manipulation, that moment takes the form

E[X λ ] = λ :|λ |=l G (l) λ,λ E[X λ ] + (♦ λ ), (2.A.25)
where, again, (♦ λ ) stands for all the term in the expansion of E[X λ ] for which the induction hypothesis, and the premise (2.A.15) of Proposition 2.A.4, provide an expansion up to the required order. Then, since I -G (l) is invertible, one can solve that joint system satisfied by the family (E[X λ ]):

E[X λ ] = (I -G (l) ) -1 (♦) λ = λ :|λ |=l (I -G (l) ) -1 λ,λ (♦ λ ). (2.A.26)
That concludes the proof of the induction step and thus the proof of the lemma.

Remark 2.A.9. The same methods as in Section 2.4 can produce the lower bound on the error Application to a 1D Ising model. The product of random matrices considered in the first sections appeared in [START_REF] Derrida | Singular behaviour of certain infinite products of random 2×2 matrices[END_REF] to express the free energy of the nearest-neighbour Ising model on the line with inhomogeneous magnetic field. The generalization considered in this appendix allows finite range interactions to be included. Let us be more precise. Consider the Ising model on T N := Z/N Z, with homogeneous interactions up to the distance d and inhomogeneous magnetic field (h k ). It is the spin model with configurations * σ ∈ {0, 1} T N whose Hamiltonian is

(-1) K+2 R K ( ) c 2(K+1) E (L X ) K+1 1 {L X B} + O( 2(K+1) ), (2.
H(σ) = k∈T N h k σ k + d l=1 α l 1 {σ k =σ k+l } . (2.A.28)
The magnetic field (h k ) k∈T N is supposed to be iid. Thanks to a transfer matrix approach, the free energy in the thermodynamic limit can be expressed through a random matrix products:

f (T ) = lim N →+∞ 1 N log Tr N n=1 A n , (2.

A.29)

where A n is a 2 d × 2 d sparse matrix (two non-zero entries on each line and each column) whose entries are the following. If τ , υ ∈ {0, 1} d , which represent the partial configuration (σ n , . . . , σ n+d-1 ) and its shift (σ n+1 , . . . , σ n+d ), then

A n (τ , υ) = exp - 1 T τ 1 h n - 1 T d l=1 α l τ l υ l 1 {τ 2 =υ 1 ,...,τ d-1 =υ d } . (2.A.30)
One can check that Assumption 2.A.1 (a) holds with N = d. Proposition 2.A.4 provides an expansion for the free energy f (T ) when the coupling constants α l tend to be very large. Set Z n = exp(-h n /T ) and l = exp(-α l /T ) for every l d. The parameters l vanish when the coupling constants α l tend to be very large. Then A n is a random perturbation of Diag(1, 0, . . . , 0, Z n ) if one writes the configurations τ , υ in lexicographic order. Thus, Proposition 2.A.4 yields

f (T ) = λ∈N d :|λ|<β c λ λ 1 1 • • • λ d d + O d l=1 β l , (2.

Part II

Pinning models with constraints Effective models for DNA denaturation. Since 1953, after the famous discovery by Francis Crick and James Watson [START_REF] Pray | Discovery of DNA structure and function: Watson and Crick[END_REF], it is known that DNA is a molecule composed of two chains -or DNA strands -that coil around each other to form a double helix. Each chain contains the genetic code used in the development, functioning and for reproduction in all known living organisms (this includes many, but not all, viruses). The two DNA strands are composed of simpler monomeric units called nucleotides. The nucleotides are almost identical and they just differ for one part that is called nucleobase, or simply base.

There are four bases: C=cytosine, G=guanine, A=adenine and T=thymine. While the nucleotides within one strand are attached to each other by strong covalent bonds, the connection between the two strands relies on the weaker hydrogen bonds between bases. The two strands, at least in living beings, are complementary according to the base pairing rules "A with T and C with G". The two strands of a double stranded DNA are said to be in their native state when all bases are paired together. This is what happens notably when the DNA is folded into the very complex structure that fits in the cell nucleus. But DNA needs to open for its two main functions: coding for proteins, and replication. Coding and replication are extremely complex processes. But since several decades researcher study in detail also the process of DNA denaturation (Figure 3.1), i.e. the unbinding of the two strands, that happens at high temperatures or that happens because a force is applied to the extremities of the strands: this second procedure can be performed thanks to the great progress that have been made in single molecule experiments. A number of models have been therefore developed to analyse these phenomena or procedures.

A very simple model, introduced by D. Poland and H. A. Scheraga [START_REF] Poland | Occurrence of a phase transition in nucleic acid models[END_REF][START_REF]Phase transitions in one dimension and the helix-coil transition in polyamino acids[END_REF][START_REF]Theory of helix-coil transitions in biopolymers[END_REF] in the 60s, has finally turned out to be a reference model, even at a quantitative level. In the Poland-Scheraga (PS) model, the double strand structure is forgotten and double stranded DNA is modeled by a linear sequence of bound pairs and loops (see Figure 3.2 in Section 3.1). The In the top figure, DNA is in its native state: all bases are paired together. In the bottom, a loop opens and DNA is denaturated at this spot. In practice -and above all approaching denaturation -due to thermal fluctuations, there are several loops even when the two strands are still together. We will talk about denaturation only when the two chains are completely detached, except at some bases at finite distance of the extremities (in fact, we will always impose that the extremities are attached, and the thermal fluctuations at the boundary lead to having some contacts of the two strands, essentially only at a finite distance from the extremities). By passing to the complementary, also the notion of native state that we will use will be somewhat generalized. loops are the regions in which the two strands are unbound. The parameters that enter the definition of the model are the energetic gain of the bound pairs and the entropic term associated to the fluctuation freedom of the loop sections. We stress that the rich DNA physical structure is reduced, in the PS model, to bond energies and loop energy+entropy term.

Aside for experimental applications, a major atout of PS model is its solvable character when one neglects that the A-T bonds are weaker than G-C bonds. With this simplification, which can viewed as looking at an homogenous version of the model, one obtains an almost explicit formula for the free energy density from which it is rather straightforward to infer the existence of a transition -the denaturation transition -along with the precise value of the critical point and the critical behaviour (Section 3.1). Such a complete solution of the homogeneous model has been crucial in obtaining then results also about the case in which one keeps the inhomogeneous nature of the original model (Section 3.2), that is, in which the A-T hydrogen bonds are weaker than G-C bonds. In physical terms, the sequence of bases can be seen as a disorder.

The Homogeneous Poland-Scheraga Model

In the beginning of the 80s there has been a wave of interest in the physical community for the exactly solvable character of the homogeneous PS model and a number of approaches, motivated both by different physical applications, and by somewhat different methods of solution. In 1984 Michael Fisher [START_REF] Fisher | Walks, walls, wetting, and melting[END_REF] observed that the solvable nature of the model can be seen in full generality in terms of a rather simple computation. In reality, as it has been pointed out in [57, App. A], Fisher's simple computation was a well-known fact in mathematics from the end of the 40s: the homogeneous PS model is just a rewriting of the basic discrete renewal process. And renewal theory greatly developed in the 50s and 60s, so the full solution of the PS model is just a corollary of known mathematics. Let us present the mathematical formalism that is commonly used for the model. This requires a short introduction on renewal processes. Discrete renewal processes. Definition 3.1.1. A process τ = (τ k ) k 0 is a discrete renewal process -its law is denoted by P -when τ 0 = 0, and the increments

(η k ) k 1 = (τ k -τ k-1 ) k 1 are IID random variables taking values in N * ∪ {∞} = {1, 2, • • • } ∪ {∞}.
We will use the notation

K(n) = P(η 1 = n): by definition, n∈N * K(n) + K(∞) = 1. Note that if η k = ∞ for one k (that is if K(∞) > 0), then τ k = τ k+1 = . . . = ∞.
In this case the renewal is called transient or terminating. Instead, if K(∞) = 0, then τ k < ∞ for all k, and the renewal is called persistent. The points (τ k ) k 0 will be called pinned points or contact points and the renewal process will often be seen as a subset τ of the discrete line N ∪ {∞}. If the renewal is terminating, then τ is a finite set and it contains the point ∞, whereas if the renewal is persistent, then τ is an infinite subset of N.

Moreover, let us recall from now the classical fundamental Renewal Theorem formula * (see e.g. [41, § XI.1])

lim n→+∞ P(n ∈ τ ) = lim n→+∞ P(there exists j such that τ j = n) = 1 E[η 1 ] ∈ [0, 1] , (3.1.1)
as soon as the renewal does not live on a sub-lattice of N (aperiodicity condition). Aperiodicity is going to be satisfied in the models we consider and, in any case, (3.1.1) is easily generalized in the periodic case. So, if τ is terminating, we have E[η 1 ] = +∞ and lim n P(n ∈ τ ) = 0, but it may happen that E[η 1 ] = +∞ also in the persistent case. This leads to the further classification into null persistent (when K(∞) = 0 and E[η 1 ] = +∞) and positive persistent (when E[η 1 ] < +∞). Note also that in the terminating and in the null persistent case, the Renewal Theorem (3.1.1) is not fully satisfactory (for one would want to know more precisely in which way the renewal function tends to zero): for this we refer to the caption of Figure 3.3. For the nomenclature, n → P(n ∈ τ ) is called renewal function.

It is an elementary fact, see e.g. [START_REF] Giacomin | Random polymer models[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF], that there is no loss of generality in assuming that K(∞) = 0 for the PS model, but we will quickly see that, nevertheless, the terminating case is going to have a central role in the arguments and in the results. In the physical literature K(n) is often chosen equal to 1/n c , up to the normalizing constant, and theoretical arguments are set forth in favor of a value of c around 2.15 for comparison with real experiments [START_REF] Blake | Statistical mechanical simulation of polymeric DNA melting with MELTSIM[END_REF], but from a theoretical viewpoint any value of c > 1 goes. We will make the more general assumption that K(•) is regularly varying with index c = 1 + α 1: there exists n 0 such that for n n 0 , We refer to [START_REF] Bingham | Regular variation[END_REF] for the many properties of slowly varying functions. We observe that the fact that K(•) is eventually positive implies that the renewal is aperiodic.

K(n) = ϕ(n) n 1+α , ( 3
A formal definition of the PS model. The PS model, corresponding to a DNA chain of length N (Figure 3.2, is introduced by modifying the distribution of the renewal via an energy reward h ∈ R for each contact point before and including N . Explicitly, the new law on the point set τ is given by

ν N,h (A) = 1 Z N,h E e h|τ ∩(0,N ]| 1 A (τ )1 N ∈τ , (3.1.4)
where A is an arbitrary collection of point set configurations, i.e. A is an element of power set of N and Z N,h is the normalization and it is usually called partition function:

Z N,h = E e h|τ ∩(0,N ]| 1 N ∈τ = N m=1 e hm P(|τ ∩ (0, N ]| = m; N ∈ τ ) = N m=1
e hm P(τ m = N ).

(3.1.5) When the contact energy h is large, the contact points are favored. Conversely, when it is very negative, the loops are favored. A competition between entropy and energy occurs and results, for N → ∞, in a phase transition between a delocalized phase, where there is a unique very large loop † and very few contact points, and a localized phase where the number of contact points is of order N and the loops are small. Everything can be made very explicit by analyzing the partition function Z N,h and this is what we explain next. The two thick lines are the DNA strands. They may be paired, gaining thus energetic contributions that depend on whether the base pair is A-T or G-C. There are then sections of unpaired bases (the loops) to which an entropy is associated: loops correspond to inter-arrival of length n ≥ 2. Now there are two possibilities: † In the technical part of the work we will prefer the mathematical terminology of big jump, with respect to large loop, but this is of course just a nomenclature issue, related to the fact that this phenomenon already appears in the random walk and renewal theory literature.

The Poland-Scheraga model

Either

∈N * e h K( ) 1 (i.e., h 0), and in this case we set K h ( ) := e h K( ) for ∈ N, and K h (∞) = 1e h 0;

Or

∈N * e h K( ) > 1 (i.e., h > 0) and in this case, by monotonicity, there exists a unique positive solution f to the equation

∈N * e h-f K( ) = 1. (3.1.7)
This defines f = f(h) and we set K h ( ) := e h-f(h) K( ) for ∈ N.

We set f(h) = 0 for h 0 so that we have K h ( ) = e h-f(h) K( ) for every h ∈ R and every ∈ N. The crucial observation is that K h (•) defines a new renewal process τ (h) that -is terminating if h < 0; -coincides with τ if h = 0 (and it may be null or positive persistent);

-is positive persistent if h > 0.
This observation is crucial because it leads to

Z N,h = exp (f(h)N ) P N ∈ τ (h) , (3.1.8)
as a direct consequence of (3.1.6) and of the fact that, for a general renewal, we have

P(N ∈ τ ) = N m=1 ∈(N * ) m : m j=1 j =N m j=1 K( j ) .
(3.1.9) Equation (3.1.8) is therefore telling us that estimating Z N,h boils down to the classical issue of estimating the renewal function; and therefore it will come with no surprise that sharp estimates on Z N,h are in the renewal process literature (see [START_REF] Caravenna | Sharp asymptotic behavior for wetting models in (1+ 1)-dimension[END_REF][START_REF] Giacomin | Random polymer models[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF] for statements and survey of the literature). Sharp estimates on Z N,h have as important consequence the full control of the path properties of the point process ν N,h in the N → ∞ limit (see Figure 3.3). But even without resorting to sharp estimates for h 0 it is very easy to see from (3.1.8) that

f(h) = lim N →+∞ 1 N log Z N,h , (3
) if h > 0. If h 0, it suffices to observe that e h K(n) P(n ∈ τ (h) ) 1.
The free energy f(•) carries already a lot of information. Let us first observe that it is a convex function (this can be seen from its definition, but it is probably more easily seen from (3.1.10) since h → log Z N,h is convex) and it is non decreasing. The analytic Implicit Function Theorem [START_REF] Krantz | A primer of real analytic functions[END_REF]Ch. 1] implies that it is real analytic for h > 0 and this is of course obvious for h < 0. Therefore h = 0 is a non analyticity point -in fact, the only one. In the statistical mechanics language, it is the critical point, and the system exhibits a phase transition at this point. The nature of this transition can also be read out of the ) that for h < 0 the density of contact sites is zero, while for h > 0 such a density is positive. But thanks to (3.1.8) one can go much beyond just by exploiting estimates on the renewal function (see [START_REF] Caravenna | Sharp asymptotic behavior for wetting models in (1+ 1)-dimension[END_REF][START_REF] Giacomin | Random polymer models[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF]). In fact ν N,h (A) can be always expressed in terms of ratios of partition functions. What one can show is that, in the limit N → ∞, the point process converges (in more than one sense, see again [START_REF] Caravenna | Sharp asymptotic behavior for wetting models in (1+ 1)-dimension[END_REF][START_REF] Giacomin | Random polymer models[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF]) to the renewal process τ (h) defined by the jump probability K h (•) (see the list after (3.1.6)). For h > 0, the Renewal Theorem formula (3.1.1) is sufficient to establish such a result, but sharp estimates on how the real function vanishes at ∞ are needed to recover such a result in the h < 0 case. For h > 0 (figure on the left), all loops are essentially of finite size (more precisely: the largest has size O(log(N )) and this logarithmic behaviour just reflects the exponentially decreasing behaviour of K h (n) for n → ∞). On the other hand, if h < 0 (figure on the right) the renewal points are finitely many and localized near the extremities: one huge loop, of size that differs from N only by a finite (random) number, occupies the whole system. At criticality, h = 0, all depends on K(•): we invite the interested reader to look at [START_REF] Caravenna | Sharp asymptotic behavior for wetting models in (1+ 1)-dimension[END_REF][START_REF] Giacomin | Random polymer models[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF]. free energy. By basic properties of convex functions, taking the derivative with respect to h can be exchanged with the limit N → ∞ in (3.1.10), as long as the f(•) is differentiable. So, if we exclude h = 0, this operation is safe and we obtain that the contact fraction coincides with f (h):

lim N →∞ 1 N E ν N,h [|τ ∩ (0, N ]] = lim N →∞ 1 N E |τ ∩ (0, N ]|e h|τ ∩(0,N ]| 1 N ∈τ E e h|τ ∩(0,N ]| 1 N ∈τ = f (h) , (3.1.11)
and therefore it is zero for h < 0 and it is positive for h > 0 (see Figure 3.4). Thus, the transition at h = 0 is between a delocalized regime in which there is zero density of contact points and a localized regime in which there is a positive density of contact points. As announced, the sharp estimates on the partition function allow to go much further and precise statements can be made that we have informally summed up into Figure 3.3 and its caption (references in the caption).

We are left with the important issue of characterizing the free energy singularity at h = 0: in physical terms this corresponds to identifying the critical behaviour (of the free energy). This just follows from an accurate analysis of n e -fn K(n) for f 0 and by applying the asymptotic inversion results for regularly varying functions [14, §1.5.7]. The final result of this rather explicit analysis is [START_REF] Giacomin | Random polymer models[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF]

f(h) h 0 ∼ c K h if τ is finite recurrent, i.e. if n nK(n) < ∞ , φα (1/h)h 1/α if τ is null recurrent, i.e. if n nK(n) = ∞ , (3.1.12)
where c K = 1/E[η 1 ] > 0 and φα (•) is a suitable slowly varying function (Figure 3.4).

The most prominent fact in (3.1.12) is that the critical exponent of the free energy is max(1, 1/α): the whole spectrum of possible singularities can be achieved by playing on the value of α. Also the case α = 0 is included, and one can in fact show that in that case the free energy in that case goes to zero faster that any power of h (but of course sharp results can be obtained, see for example [START_REF] Alexander | Equality of critical points for polymer depinning transitions with loop exponent one[END_REF]).

0 h f(h) (a) α < 1. 0 h f(h) (b) α > 1.

Figure 3.4:

Free energy h → f(h) and its critical behaviour: when α > 1, the transition is first order (∂ h f(h) is discontinuous at h = 0) whereas it is at least second order when α < 1.

Remark 3.1.3. We observe that in the physical literature we speak of transition of order k = 1, 2, . . . if the free energy is C k-1 , but not C k , at criticality. Therefore (3.1.12) strongly suggests that the order of the transition in the PS model is given by the integer part of 1/α (see [START_REF]Disorder and critical phenomena through basic probability models[END_REF]Sec. A.2.3] for a proof). Moreover, a transition is of infinite order if the free energy is C ∞ , but (of course) not analytical, and, for the PS model, the free energy vanishes faster than any power of h minus its critical value (which is zero in this specific case).

The Disordered Poland-Scheraga Model

If we want to use the PS model for modeling real DNA denaturation we would need to use a model whose partition function is for example (with T the temperature)

E e -1 T N j=1 e j δ j 1 N ∈τ , (3.2.1)
where δ j = 1 j∈τ and e j are the (negative) energies that correspond to the A-T and G-C bonds: we have therefore a deterministic inhomogeneous sequence (e j ), with entries that take two values. In reality, the situation is definitely more complex than this and some generalized models that take into account the possibility of mismatches or of non perfect complementarity of the two DNA strands have been used in applications and studied theoretically (see Section 3.3). But even sticking to the very basic model defined by (3.2.1), it is not easy to extract mathematical results for given deterministic sequences. The theoretical research (not only in mathematics, but also in physics) has then focused on the case in which {e j } j=1,2,... is the realization of a sequence of random variables and, in most of the cases, these random variable are IID. We use the notation

Z h,ω,β N = E e N j=1 (βω j +h)δ j 1 N ∈τ , (3.2.2)
which just corresponds to -e j /T = βω j + h: {ω j } j∈N * is a sequence of IID random variables, independent of τ and whose law is denoted by P, on which we assume E[exp(tω 1 )] < ∞ for every t ∈ R. Moreover, without loss of generality we can and do assume that

E[ω 1 ] = 0 and E[ω 2
1 ] = 1 because we have the two parameters β 0 that plays the role of standard deviation and h ∈ R that plays the role of the mean-value.

Remark that now Z h,ω,β N is a random variable: we define the quenched free energy as

f(β, h) := lim N →∞ 1 N E log Z h,ω,β N . (3.2.
3)

The existence of this limit follows by standard super-additive arguments. We can of course apply these arguments also to the non disordered case (f(0, h) = f(h)!), but with β = 0 we had a rather explicit solution that is no longer available. Nevertheless (β, h) → log Z N,ω,β,h is still convex, therefore so is (β, h) → f(β, h), and the function h → f(β, h) is non decreasing. Moreover Proposition 3.2.1. For every β > 0 we have that

f(0, h) f(β, h) f(β, h + λ(β)) , (3.2.4
)

with λ(β) = log E[exp(βω 1 )].
Proof. The upper bound, called annealed bound, is a consequence of Jensen's inequality:

E log Z h,ω,β N log E[Z h,ω,β N ] = log Z N,h+λ(β) . (3.2.5)
The lower bound is a consequence of the monotonicity of β → f(β, h) for β 0 (which follows from convexity in β and from the fact that

∂ β E log Z h,ω,β N is zero at β = 0).
Therefore the graph of h → f(β, h) looks like as explained in Figure 3.5 and, like for the β = 0 case, we necessarily have a transition at

h c (β) := inf{h ∈ R : f(β, h) > 0} . (3.2.6)
By the very same arguments as for the β = 0 case, we realize that we are dealing again with a transition between a delocalized regime and a localized one. Remark also that a direct consequence of Proposition 3.2.1 is that

-λ(β) h c (β) 0 , (3.2.7)
and in full generality one can show that the second inequality is strict: h c (β) < 0 (see [START_REF] Giacomin | Random polymer models[END_REF]Th. 5.2] and [START_REF] Hollander | Random polymers[END_REF]Ch. 7]). The first inequality is in some cases an equality, as we will see (cf. Theorem 3.2.2). We refer to [START_REF]Disorder and critical phenomena through basic probability models[END_REF] for a review on the literature on the several results on the trajectories of the model that go beyond what one obtains by differentiating the free energy. Here we focus on the issue of whether or not the disorder affects the critical behaviour. Since the solvability of the model is lost, a priori this looks like a daunting question (and in fact, to a certain extent, it is still very open). There is nevertheless a remarkable approach in physics, due to A. B. Harris [START_REF] Harris | Effect of random defects on the critical behaviour of Ising models[END_REF], and based on the renormalization group approach (Harris criterion) that, for the PS model [START_REF] Derrida | Effect of disorder on two-dimensional wetting[END_REF][START_REF] Forgacs | Exact critical behaviour of two-dimensional wetting problems with quenched disorder[END_REF], leads to the prediction that -if α < 1/2 then, at least for β not too large, disorder is irrelevant which, in general, means that the critical exponent of the β > 0 case -more precisely β ∈ (0, β 0 ) for some β 0 > 0 -is the same as for the β = 0 case. For the PS model the prediction is also that h c (β) = -λ(β), if disorder is irrelevant.

0 h f(β, h) h c (β)
-if α > 1/2 disorder is relevant, so the critical exponent is predicted to be different from the β = 0 case for every β > 0. Moreover h c (β) is predicted to be larger than -λ(β).

The Harris criterion is normally stated in the elegant and compact form:

disorder is irrelevant if νd > 2 , (3.2.8)
where ν is the critical exponent of the correlation length of the non disordered system and d is the dimension. There are several ways to define the correlation length in a statistical mechanics system, but any reasonable definition should yield the same critical exponent (at least for non disordered systems [START_REF] Chayes | Correlation length bounds for disordered Ising ferromagnets[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF]). For example, for the PS model we can take as the correlation between sites j and k simply the correlation between the random variables δ j and δ k in the N → ∞ limit and one can show that such a correlation decays for |j -k| → ∞ like exp(-|j -k|/L β (h)) [START_REF]Renewal convergence rates and correlation decay for homogeneous pinning models[END_REF][START_REF]The localized phase of disordered copolymers with adsorption[END_REF][START_REF] Toninelli | Correlation lengths for random polymer models and for some renewal sequences[END_REF] for a certain quantity L β (h). This L β (h) is therefore a natural correlation length and log

L β (h) ∼ ν β log(1/(h -h c (β)) for h h c (β).
For pinning models it turns out that log L β (h) ∼ log(1/f(β, h)) approaching criticality, see for example [START_REF] Derrida | Effect of disorder on two-dimensional wetting[END_REF][START_REF] Forgacs | Exact critical behaviour of two-dimensional wetting problems with quenched disorder[END_REF] for the physical prediction and [START_REF]Renewal convergence rates and correlation decay for homogeneous pinning models[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF][START_REF]The localized phase of disordered copolymers with adsorption[END_REF][START_REF] Toninelli | Correlation lengths for random polymer models and for some renewal sequences[END_REF] for proofs. Therefore ν = ν 0 = max(1, 1/α) and, since d = 1, (3.2.8) becomes max(1, 1/α) > 2, that is α < 1/2. Harris' arguments are essentially a linear stability analysis for the renormalization transformation: νd > 2 is the stability condition for the non disordered model, and the renormalization flow is supposed to damp the disorder on large scale (irrelevant disorder). The renormalization group is instead unstable for max(1, 1/α) > 2 and there is no reason to believe that the large scale behaviour of the disordered system has anything to do with the non disorder system behaviour (relevant disorder). But Harris criterion predicts nothing about the critical behaviour of the disordered system when disorder is relevant.

Mathematically results go very far for the role of disorder in PS model. In particular, the irrelevant disorder regime is fully under control and disorder relevance is rigorously established. We sum up the most relevant results: Theorem 3.2.2. For every β > 0 the map h → f(β, h) is C ∞ on (h c (β), +∞) [START_REF]The localized phase of disordered copolymers with adsorption[END_REF] (we have already seen that it is convex and increasing). Moreover

1. if α < 1/2 then h c (β) = -λ(β) and f(β, h c (β) + ) 0 ∼ f(0, ) 0 ∼ 1 α φ(1/ ) , (3.2.9)
for a suitable slowly varying function φ(•) [START_REF] Alexander | The effect of disorder on polymer depinning transitions[END_REF][START_REF]On the irrelevant disorder regime of pinning models[END_REF][START_REF] Lacoin | The martingale approach to disorder irrelevance for pinning models[END_REF][START_REF]A replica-coupling approach to disordered pinning models[END_REF];

2. if α > 1/2 then h c (β) > -λ(β) and thanks to the smoothing bound [START_REF] Caravenna | A general smoothing inequality for disordered polymers[END_REF][START_REF]Smoothing effect of quenched disorder on polymer depinning transitions[END_REF][START_REF]Smoothing of depinning transitions for directed polymers with quenched disorder[END_REF] f(β, h c (β) + )

0 = O( 2 ) , (3.2.10)
that holds for every α, we see that the critical behaviour for β > 0 is different than the critical behaviour for β = 0 if α > 1/2:

lim inf 0 log f(β, h c (β) + ) log 2 > max 1, 1 α . (3.2.11)
Moreover (3.2.10) implies that ∂ h f(β, •) is continuous on R for every value of α.
The delicate marginal disorder case is treated in [START_REF] Berger | Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift[END_REF][START_REF] Giacomin | Marginal relevance of disorder for pinning models[END_REF] to which we refer also for the related literature. In particular, the slowly varying function ϕ(•) is crucial to determine whether disorder is relevant or irrelevant.

Many questions concerning the complete behaviour of the disordered PS free energy are still open, in particular:

-is f(β, h) analytic for h > h c (β) (and, of course, β > 0)? -when disorder is relevant, in particular therefore for α > 1/2, it is not known if a critical exponent γ exists and, of course, we do not know its value (as we have seen, the smoothing bound implies γ 2). The predictions in the physical literature are not fully consistent, but the prediction γ = ∞ has been set forth (see [START_REF]Disorder and critical phenomena through basic probability models[END_REF]Ch. 5] for a review of the literature on this issue) and recently this has been proven for two very special simplifications/variations of the PS model [START_REF] Berger | Disorder and critical phenomena: the α = 0 copolymer model[END_REF][START_REF] Chen | In preparation[END_REF][START_REF] Derrida | The depinning transition in presence of disorder: a toy model[END_REF] We end this section by signaling also the Large Deviations approach to PS models and to more general disordered pinning models developed in [START_REF] Birkner | Quenched large deviation principle for words in a letter sequence[END_REF][START_REF] Bolthausen | Random copolymers, Correlated random systems: five different methods[END_REF][START_REF] Bolthausen | A copolymer near a selective interface: variational characterization of the free energy[END_REF] and references therein.

Beyond the Poland-Scheraga model

Generalized Poland-Scheraga. It is impossible to give here a proper account of the several models that go toward a more realistic, or at least more detailed, modeling of the DNA structure. But one direction, that we have already briefly mentioned, is the one of trying to account for mismatches and slips in the matching of the two strands, or to account for non exact complementary of the two strands, or even possibly for different length of the two strands. The natural generalization of the PS model, proposed in [START_REF] Garel | Generalized Poland-Scheraga model for DNA hybridization[END_REF][START_REF] Neher | Intermediate phase in DNA melting[END_REF] and mathematically studied in [START_REF] Berger | DNA melting structures in the generalized Poland-Scheraga model[END_REF][START_REF] Giacomin | Generalized Poland-Scheraga denaturation model and twodimensional renewal processes[END_REF], accounts for the richness of the possible DNA structures that we just outlined and, very remarkably, keeps the solvable character of the homogeneous version of the model. This new model is called Generalized Poland-Scheraga (GPS) model. We refer to [START_REF] Giacomin | Generalized Poland-Scheraga denaturation model and twodimensional renewal processes[END_REF] for a mathematical introduction to this model and we just quickly mention that the solvable nature is once again connected to a renewal structure embodied in the model, but this time the renewal is two dimensional. The GPS model is of interest to us because of its richer phenomenology: we will find it also in the models that we will consider. More precisely, the GPS has a localization/delocalization transition that corresponds to DNA denaturation, but it has also other transitions that happen in the localized regime, that is when the two strands are in their native state. We stress from now that these new transitions have been identified rigorously only in absence of disorder, so, till specified otherwise, we limit ourselves to the non disordered case. What happens in the GPS model if the two strands have different lengths M > N (and if the strands are bound together at both extremities: we make this choice for ease of exposition, see [START_REF] Berger | DNA melting structures in the generalized Poland-Scheraga model[END_REF][START_REF] Giacomin | Generalized Poland-Scheraga denaturation model and twodimensional renewal processes[END_REF] for the general case) is that they can bind together in two different ways:

1. the two strands fully bind together (Figure 3.6) in the sense that essentially all loops have finite length (more precisely, the largest loop is of length O(log N ) for N → ∞): this is possible because asymmetric loops are allowed in the GPS model, that is loops that involve a different number of bases for the two stands;

2. the two strands bind together in such a way that essentially all the M -N bases that are in excess are absorbed by one large loop (Figure 3.7): again, this is possible because asymmetric loops are allowed. This appearance of a large loop has been presented in then physical literature as a condensation phenomenon. In [START_REF] Berger | DNA melting structures in the generalized Poland-Scheraga model[END_REF][START_REF] Giacomin | Generalized Poland-Scheraga denaturation model and twodimensional renewal processes[END_REF] the transition between localization without a large loop and with a large loop is viewed from Large Deviations perspective as the passage from Cramer Deviations (i.e. deviations that can be captured by an exponential tilt of the measure with which one modifies the expectation of the underlying jumps) to non Cramer Deviations, in which no exponential tilt of the measure can lead to an expectation of the underlying jumps compatible with the constraint that the two chains meet at the extremities.

The study of this phenomenon, even beyond Large Deviations and the Cramer versus non Cramer behaviour, is already present in the mathematical literature on random walks and renewal processes and the keyword is: big jump regime.

A mathematical analysis of the denaturation transition in the disordered case has been developed in [START_REF]Disorder and denaturation transition in the generalized Poland-Scheraga model[END_REF]13] and no work has been done on the effect of disorder on the large loop transitions (one can find some numerics in [START_REF] Neher | Intermediate phase in DNA melting[END_REF]).

A model for circular DNA. The generalization of the PS model we are interested in goes a priori toward a different direction. One physical aspect of DNA that is taken into account only in a very indirect way both in the PS and GPS models is the fact that the helix structure involves a global rotation effect when a loop opens. This is treated as a minor effect in the PS and GPS models because the helix is free to rotate. But DNA appears in nature also in a different geometry for which one cannot neglect the effect we just mentioned: the circular geometry. We talk about circular DNA when each of the two DNA strands close, thus forming two rings, entangled by their coiling around each other. This is not an exceptional geometry: it is the standard one for the so called plasmids that are very actively studied for several reasons, notably their involvement in important pathologies and their use as vehicle of genetic code in genetical engineering, see for example Plasmids on the Scitable of Nature Education (www.nature.com/scitable). So, in a double stranded DNA with a ring structure, one can no longer neglect the winding of the chain: one can easily set up a macroscopic experiment with two pieces of rope and see what happens when one tries to separate the two ropes.

Here is a very informal view of what happens in circular double stranded DNA:

-the two strands can no longer get completely separated;

-if a loop opens, then the winding lost in that area (that can be quantified in a number) must be absorbed somewhere else in the chain, for example in the backbone of the two strands (with an energetic cost) or in some regions in which the two strands curl around each other and bend in an atypical way (again with an energetic cost).

Our purpose is to tackle mathematically the models proposed in [START_REF] Bar | Denaturation of circular DNA: Supercoil mechanism[END_REF][START_REF]Denaturation of circular DNA: Supercoils and overtwist[END_REF] for circular DNA: the variety corresponds to different mechanisms for absorbing the excess winding. For all of them, it is pointed out that there can be a condensation phenomenon like for the GPS model: the two strands cannot fully separate, but it might happen that a large loop, i.e. of a size proportional to the length of the chains, appears. We will present in Chapter 4 a general framework that -includes the circular DNA models of [START_REF] Bar | Denaturation of circular DNA: Supercoil mechanism[END_REF][START_REF]Denaturation of circular DNA: Supercoils and overtwist[END_REF], but it includes also models in which there can be also a denaturation transition;

-can be seen as a toy model for the GPS, and for transitions of the type large loop in the localized regime. The reason is simply that the large loop phenomenon emerges because of the same mechanism: the big jump one. In this general framework we will give a full analysis of the homogeneous models: this includes a complete understanding of the transitions and critical phenomena, along with precise results on the behaviour of the trajectories of the polymer in all the regimes, including the critical one.

But we will also tackle the disordered case (Chapter 5) and we will present results that, unlike what is predicted in the physical literature [7, § IV], show that disorder suppresses the large loop/big jump phenomenon.

Introduction

As we have seen in the previous chapter, generalizations of the Poland-Scheraga (PS) model have been proposed to go beyond the rigid constraints of the PS model, notably the exact symmetry of the two strands, and to allow for richer geometrical structures (see 79 Section 3.3). In this chapter, we focus on models of circular DNA, that is cases in which the DNA consists of two rings, entangled by their coiling around each other. In the end, the annulus structure turns out to be a real obstacle to a full denaturation transition: unlike the PS model (in which the two strands are free to rotate), in circular DNA the linking number must be conserved. Thus, a loop opening creates an excess of winding somewhere else in the chains. In particular the two strands cannot fully separate. Various models have been proposed in the biophysical literature for circular DNA corresponding to different mechanisms for absorbing the excess winding. We focus on the biophysics works [START_REF] Bar | Denaturation of circular DNA: Supercoil mechanism[END_REF][START_REF]Denaturation of circular DNA: Supercoils and overtwist[END_REF] and we propose a general mathematical model (or class of models), based on discrete renewal processes, that includes all the models in [START_REF] Bar | Denaturation of circular DNA: Supercoil mechanism[END_REF][START_REF]Denaturation of circular DNA: Supercoils and overtwist[END_REF]. In this framework, the large loop phenomenon, predicted by [START_REF] Bar | Denaturation of circular DNA: Supercoil mechanism[END_REF][START_REF]Denaturation of circular DNA: Supercoils and overtwist[END_REF], is directly related to the big jump phenomenon for the renewal and the general model we present can also be seen as toy model for the GPS model [START_REF] Berger | DNA melting structures in the generalized Poland-Scheraga model[END_REF][START_REF]Disorder and denaturation transition in the generalized Poland-Scheraga model[END_REF][START_REF] Giacomin | Generalized Poland-Scheraga denaturation model and twodimensional renewal processes[END_REF] mentioned in Section 3.3.

Circular DNA model(s)

Let us start by laying out the model proposed by Bar, Kabakçıoğlu and Mukamel [START_REF] Bar | Denaturation of circular DNA: Supercoil mechanism[END_REF][START_REF]Denaturation of circular DNA: Supercoils and overtwist[END_REF].

It is an homogeneous model as it neglects that the A-T bonds in the strand pairing are weaker than the G-C bonds. A double stranded circular DNA of length N is modeled by a sequence of bond segments, supercoil segments and loops. The contribution of these three types of segments to the energy and to the entropy are the following. (ii) A supercoil segment of length k forms with probability proportional to e βkes = ω k s . These internal energies typically satisfy e b < e s < 0.

(iii) A loop of length k forms with probability proportional to

ω k k c .
A supercoil segment corresponds to area where the DNA double helix tangles around itself (as the cord on landline telephones sometimes does). Denote by L b , L s and L the total length of bonds, supercoils and loops (L b is the sum of the lengths of all bond segments, etc.). Three mechanisms have mainly been considered in the biophysics literature to model the winding constraint in circular DNA.

1. In the first model, the excess of winding created by a loop opening is compensated by supercoils and an overtwist, somewhere else in the chain. The supercoils can exactly compensate for the open loops, but if not, the excess linking number L -L s causes an overtwist, supported by the L b + L s contact points between the two strands of the DNA. The average stacking angle per contact point is ∆θ = (L -L s )/(L b + L s ) and the overtwist induces an energy penalty per contact point proportional to (∆θ) 2 , say χ • (∆θ) 2 . The total energy penalty due to overtwisting is therefore

(L b + L s ) • χ • (∆θ) 2 = χ (L -L s ) 2 L b + L s . (4.1.1)
This model will be referred to as the model with overtwist and supercoils. Two other models has also been considered in the literature, which are actually degenerate cases of this first one.

2. In the second model, the supercoil phenomenon does not exist (ω s = 0 and L s = 0). The energy cost of overtwisting is then χL 2 /L b . This model will be called the overtwist model.

3. In the third model, instead of an energy cost χ • (∆θ) 2 , the winding constraint is rigid and no overtwist is allowed. It can be recovered by taking χ = ∞ in the first model. This means that only the configurations in which L s = L are possible. It will be referred to as the model with strong constraint and supercoils.

For these models a large loop transition is predicted in the physics literature [START_REF] Bar | Denaturation of circular DNA: Supercoil mechanism[END_REF][START_REF]Denaturation of circular DNA: Supercoils and overtwist[END_REF]. At low temperature, full localization is observed: bonds, loops and supercoils (except in the second model, of course) coexist (the lengths L b , L s , L are of order N ) but the loops are typically small. Above a critical temperature, a macroscopic (i.e., of size proportional to N ) loop forms, but, unlike a delocalized phase, the two strands are not completely separated: the density of contact points stays positive (the lengths L b and L s are also of order N ): this is the big jump phase.

We stress that the big jump phenomenon -also called condensation phenomenon or non Cramer Large Deviations regime in different contexts discussed in the previous chapter -is not delocalization and the transition is not a localization/delocalization transition: this will be seen notably in the different critical behaviour and it will become even more evident in the next chapter where the disordered version of the model is introduced. Nevertheless, the general framework we are going to study can and does accommodate localization/delocalization transitions, providing therefore a full analogy with the GPS model of Section 3.3. We will however minimize the analysis of this aspect of the model, for sake of clarity of the presentation.

A mathematical model

From a mathematical perspective, the mechanisms involved in circular DNA will be studied using a modification of the Poland-Scheraga (PS) model. In this model, the sequence of loops is represented by a discrete renewal process (see Section 3.1). We recall the main definitions and notations. Definition 4.1.1. A process τ = (τ k ) k 0 is a discrete renewal process under distribution P when τ 0 = 0, and the increments (η k ) k 1 = (τ kτ k-1 ) k 1 are IID random variables taking values in N * = {1, 2, • • • }. We will use the notation K(n) = P(η 1 = n). The points (τ k ) k 0 will be called pinned points or contact points and the renewal process will often be seen as a subset of the discrete line N. For convenience, we use a slightly different notation with respect to Chapter 3: when the length N of the DNA strands is fixed, the number of contact points is

|τ | = sup{k 1 | τ k N }, (4.1.2)
it plays the role of what was denoted by |τ ∩ (0, N ]| in Chapter 3. Similarly, the set of contact points is restricted to (0, N ]:

τ = {τ k : 1 k |τ |} ⊂ (0, N ]. (4.1.3)
In that formalism, the loops are the segment between two contact points which are not immediately adjacent in the DNA strand, i.e., the segments for which τ kτ k-1 2. The total length of loops is

L = N -|τ |. (4.1.4)
In the PS model, the distribution of the renewal is modified by giving an energy reward h ∈ R for each contact point, yielding a new distribution (see Section 3.1)

ν N,h (A) = 1 Z N,h E e h|τ | 1 A (τ )1 N ∈τ , (4.1.5)
where A is an arbitrary collection of point set configurations, i.e. A is an element of power set of (0, N ]. Formally, ν N,h is a measure on the power set of ((0, N ]), equipped with its discrete σ-algebra F .

To model bonds and supercoils, consider, independently of the renewal, IID fair Bernoulli random variables v 1 , . . . , v N taking value in {b, s}:

that is P(v 1 = b) = P(v 1 = s) = 1/2.
The contact points tagged with b are the bonds and the contact points tagged with s are the supercoils. The sets of bonds and supercoils are respectively

B = {j ∈ τ | v j = b}, S = {j ∈ τ | v j = s} (4.1.6)
and their cardinals

L b = |B| = j∈τ 1 v j =b , L s = |S| = j∈τ 1 v j =s , (4.1.7)
are the total length of bonds and the total length of supercoils. Note that B S = τ and L b + L s = |τ |. Before explaining our general framework, we detail what the first model becomes in that mathematical formalism.

An example: the model with supercoils and overtwisting. We assign a reward h for each contact point and a reward w for each supercoil. The Boltzmann weight associated to a configuration τ is

exp h(L b + L s ) + wL s -χ (L -L s ) 2 L b + L s . (4.1.8)
Therefore, the distribution of the renewal is modified in the following way: for each A ∈ F ,

ξ os,h N (A) = 1 Z os,h N E exp h(L b + L s ) + wL s -χ (L -L s ) 2 L b + L s 1 A (τ )1 N ∈τ . (4.1.9)
If we decompose this expectation depending on the numbers m = |τ | and p = L s , we get

E exp h(L b + L s ) + wL s -χ (L -L s ) 2 L b + L s 1 A 1 N ∈τ = N m=1 m p=0 E exp hm + pw -χ (N -m -p) 2 m 1 A 1 τm=N 1 Ls=p = N m=1 m p=0 exp hm + pw -χ (N -m -p) 2 m P(A; τ m = N )P(Bin(m, 1/2) = p) = N m=1 e hm Ψ os (m, N )P(A; τ m = N ) = E e h|τ | Ψ os (|τ |, N )1 A 1 N ∈τ , (4.1.10)
where

Ψ os (m, N ) = m p=0 exp pw -χ (N -m -p) 2 m 2 -m m p . (4.1.11)
The partition function can also be rewritten

Z os,h N = E e h|τ | Ψ os (|τ |, N )1 N ∈τ = N m=1 e hm Ψ os (m, N )P(τ m = N ). (4.1.12)
The same type of decomposition holds for the PS model (3.1.5). But here the energy factor e hm is replaced by a more complex factor e hm Ψ os (m, N ) which takes into account the geometric constraint of circular DNA. The same structure emerges for the two other models detailed in the biophysics introduction and a general convolution formalism can be set up to include all of them.

General framework. The next definition provides a general framework to study geometric mechanisms such as those involved in circular DNA. It takes the form of a slightly more general modification of the renewal distribution:

ξ Ψ,h N (A) = 1 Z Ψ,h N E e h|τ | Ψ(|τ |, N )1 A 1 N ∈τ (4.1.13)
First observe that the measure and the partition function can be decomposed depending on the number of contact points:

Z Ψ,h N = N m=1 E e h|τ | Ψ(|τ |, N )1 |τ |=m 1 N ∈τ = N m=1
e hm Ψ(m, N )P(τ m = N ). (4. 1.14) The competition between the energy factor e hm and the entropy P(τ m = N ) which occurs in the PS model is replaced by a competition between a more complex factor e hm Ψ(m, N ), and the entropy P(τ m = N ). These new models appear as a convolution between a PS model with a fixed number of contact points, and a nonnegative kernel Ψ representing the geometric mechanism. That's why these models will be called kernel models. The required properties on the kernel Ψ will be detailed later. Before, let us consider again the biophysics models of the introduction.

Example 4.1.2. The three biophysic models introduced page 80 (items 1, 2 and 3) to model circular DNA can be written and studied in this general framework. For each one we give the corresponding measure, for A ∈ F , and the kernel Ψ.

(Overtwist and supercoils)

ξ Ψos,h

N (A) = 1 Z Ψos,h N E exp h(L b + L s ) + wL s -χ (L -L s ) 2 L b + L s 1 A (τ )1 N ∈τ , Ψ os (m, N ) = m p=0 exp pw -χ (N -m -p) 2 m 2 -m m p .
(4.1.15) Note that if χ = 0, then it boils down to a (shifted) standard Poland-Scheraga model: Ψ os (m, N ) = e mh 0 with h 0 = log(e w + 1)log 2.

(Overtwist) ξ

Ψo,h N (A) = 1 Z Ψo,h N E exp hL b -χ L 2 L b 1 A (τ )1 N ∈τ , Ψ o (m, N ) = exp -χ (N -m) 2 m . (4.1.16) 3.
(Strong constraint and supercoils) It is the limit χ → +∞ of the model with overtwist and supercoils:

ξ Ψs,h N (A) = 1 Z Ψs,h N E e h(L b +Ls) 1 A (τ )1 N ∈τ 1 L =Ls , Ψ s (m, N ) = 2 -m m N -m 1 N 2m . (4.1.17)
The big jump transition described in the physics literature [START_REF] Bar | Denaturation of circular DNA: Supercoil mechanism[END_REF][START_REF]Denaturation of circular DNA: Supercoils and overtwist[END_REF] and the corresponding singularity of the free energy will be recovered, under some hypothesis on the distribution of the renewal. For the circular DNA models it is the only phase transition. But, in addition, certain classes of kernels Ψ (without physical interpretation if we restrict to [START_REF] Bar | Denaturation of circular DNA: Supercoil mechanism[END_REF][START_REF]Denaturation of circular DNA: Supercoils and overtwist[END_REF]) can yield other types of phase transitions, including a delocalization transition if Ψ( N, N ) is smooth around = 0. Those will be discussed in subsection 4.2.4.

Given the convolution representation (4.1.14), a first step to tackle these models is to study the PS model with a given number of contact points. Not only will it be useful to compute their partition function and their free energy but also to understand these new models from a trajectory viewpoint.

For 1 L N , we consider the measure

µ N,L (A) = P τ ∈ A N ∈ τ ; |τ | = L = P τ ∈ A τ L = N . (4.1.18)
The associated partition function is P(τ L = N ). With a tiny change of viewpoint, µ N,L is also the equilibrium measure of an irreducible zero-range process with L sites and N particles (see [START_REF] Armendáriz | Thermodynamic limit for the invariant measures in supercritical zero range processes[END_REF]). In that framework, many path properties of this model have been investigated, which we will translate in our setting. This model will be called constrained PS model, or PS model with a fixed density, or a fixed number of contact points.

Notations and Main Assumptions

-The letters c, C, ... will denote positive constants, whose value does not matter and can change from one line to another.

-Similarly, φ will be a generic slowly varying function (see [14, §1.2.1]) which can change from one line to another, whereas ϕ will be the one defining K (see (4.1.19)).

-N , L will be two integers, with L N .

About the renewal. The hypotheses on the renewal τ are the same as the usual assumptions made for the study of the PS model (see Chapter 3). Though, for convenience, we will use slightly different notations.

Assumptions 4.1.3 (on the renewal distribution). The renewal distribution K regularly varies with n: there exists α 0 and a slowly varying function ϕ such that

K(n) = ϕ(n) n 1+α , n 1. (4.1.19)
We also suppose K(1) > 0.

We define K(u) for u ∈ (0, +∞) with the same formula. Set

E[η] = n nK(n) ∈ (1, +∞], c = 1 E[η] ∈ [0, 1), σ 2 = Var(η). (4.1.20) Of course E[η] = +∞ if α < 1 and it is finite if α > 1.
As in the study of the PS model, we will need to consider tilted measures. For x 0, consider the measure P x under which the renewal has distribution

K x (n) = K(n)e -xn E[e -xη ] . (4.1.21)
The corresponding variance is

σ 2 x = Var x (η) = E x [η 2 ] -E x [η] 2 . ( 4.1.22)
We will also need to consider, for any N LE[η], the unique x L 0 (which actually also depends on N ) such that (ii) (a) For every ∈ (a, b), Note that these assumptions imply that for any ∈ (a, b),

E x L [η] = N L . ( 4
sup |m-N | √ N log N Q(m, N ) Q( N , N ) -1 -→ N →+∞ 0. ( 4
c 1 N -c 1 Q(m, N ) c 2 N c 2 , for m N ∈ [u, v]. ( 4
1 N log Ψ(m, N ) -→ N →+∞ m N → H( ). (4.1.28)
These assumptions are fulfilled by the three models introduced before. A formula can be given for the function H for both of them. Actually for none of them the term e -BN is needed. In fact, the term e -BN is only useful for some models for which, for instance, a > 0 (and H( ) = -∞ for < a) but Ψ(N, m) is not exactly zero for m/N < a (it is just very small). 1}. Thus, H os is also concave. Moreover H os is analytic on (0, 1), with H os (0) = -∞.

(Overtwist)

H o ( ) = -χ (1 -) 2 , Q o (m, N ) = 1. ( 4
H s ( ) = -log 2 + log -(1 -) log(1 -) -(2ρ -1) log(2ρ -1) if ∈ 1 2 , 1 , -∞ if ∈ 0, 1 2 , (4.1.33) and Q s (m, N ) ∼ m 2π(N -m)(2m -N ) . ( 4

.1.34)

If α = 1 then g vanishes faster than any power of (c ):

log |g( )| log( -c ) -→ c +∞, (4.2.4)
and formally, κ = +∞.

Free energy for the kernel models

We start by the definition the free energy f H , which is naturally expressed with a variational formula. Lemma 4.2.4. For any h ∈ R,

1 N log Z Ψ,h N -→ N →+∞ f H (h) = sup ∈[0,1] { h + H( ) + g( )} . (4.2.5)
Now we discuss the regularity and the singular behaviour of the free energy f H . Recall the conventions of Assumptions 4.1.4 for H . First, and most importantly, the singularity of g at c leads to a singularity at point

h c := -H ( c ) (4.2.6)
for the kernel model free energy f H . Depending on H, this point may not exist (i.e., be infinite): see Example 4.2.5 for the three circular DNA models. An important result, we will see, is that the singular behaviour of f H at h c is precisely the same as the singular behaviour of g at c . In particular the critical exponent is also κ. In fact, the corresponding phase transition on trajectories is the same: it is the big jump/localization phase transition, which kernel models will also exhibit (see the next subsection). For the three biophysical models of Example 4.1.2, it is the only singular point. But for some smoother kernels (with no physical significance, e.g., H(x) = x(1x)), there can be two other critical points

h = -H (a) -g (a), h = -H (b) -g (b), (4.2.7)
where f H exhibits a singular behaviour. Once again, depending on H, these points may not exist (i.e., be infinite), and the actual set of critical points, in all generality, is

C = {h c , h , h } ∩ R. ( 4
) If E[η] < +∞ then h c ∈ R; otherwise h c = -∞. 2. (Overtwist) If E[η] < +∞ then h c = -χ(1 -2 c )/ 2 c ; otherwise h c = -∞.

(Strong constraint and supercoils

) If E[η] < 2 then h c = log 2(2 c-1) 2 c(1-c) ; otherwise h c = -∞.
For the sake of completeness, the proofs will be written in all generality. But, at first reading, and for the applications, one may forget the points h and h , or suppose h = -∞ and h = +∞. The first result states that h c , h and h are indeed the only possible critical points. Proposition 4.2.6. The free energy h → f H (h) is convex and continuously differentiable on R and it is analytic on R\C . Remark 4.2.7. A quick study will tell that f H is linear for h h and h h with slopes a and b, and that f

H (h) ∈ (a, b) for h ∈ (h , h ). The critical point h c is precisely the solution of f H (h) = c .
Here is the main result, stating, in a nutshell, that the critical exponent at h c is κ. Proposition 4.2.8 (Singular behaviour of f H : big jump transition). Suppose a < c < b. Then the function f H exhibits at h c the same singularity as g at c . More precisely, set

(f H ) reg (h) = sup ∈[0,1] { h + H( )} , (4.2.9)
which is analytic on (h , h ), and f

H (h) = (f H ) reg (h) for h h c . Then, if α = 1, there exists c > 0 such that f H (h) -(f H ) reg (h) ∼ h hc c g c h h c ∼ h hc -(h -h c ) κ φ((h -h c ) -1 ); (4.2.10
)

and if α = 1, lim h hc log(f H (h) -(f H ) reg (h)) log(h -h c ) = +∞. (4.2.11)
This result motivates further remarks.

Remark 4.2.9.

-Since f H is non-trivial on both the left side and the right side of h c , one needs to subtract a regular part (here the left part) to exhibit the singularity.

-The constant c in (4.2.10) can be made explicit and be expressed with the parameters of the model (h c , c , κ, H ( c ) and σ 2 ) (see the proof).

-When α = 1, the critical exponents are formally +∞ for both f H and g, but the asymptotic equivalence (4.2.10) is not necessarily recovered.

-The singular behaviour when h approaches h or h (when they are finite) will be briefly discussed in Subsection 4.2.4.

It is interesting to compare with the PS free energy

f(h) = lim N →+∞ 1 N log Z N,h = sup ∈[0,1]
{ h + g( )}, (4.2.12)

for which the unique phase transition occurs at h c = 0, and the critical exponent is max(1, 1/α) (see (3.1.12)). In fact, this can be recovered in our setting, by using the variational formula (4.2.12) and the properties of g.

Path Properties for the kernel models

In this subsection, we state formally the results about the big jump phase. The relevant indicator is the size of the largest jump of the renewal

M |τ | = max 1 k |τ | η k = max 1 k |τ | (τ k -τ k-1 ), (4.2.13)
under the measure ξ Ψ,h N and then under the measure µ N,L . Depending if the density is smaller or larger than c , it will behave drastically differently. In some cases, looking at the size of the second largest jump,

M |τ | = max 1 k 1 =k 2 |τ | min(η k 1 , η k 2 ), (4.2.14)
can be relevant. Here are the main results concerning the kernel models path properties. We recall * that for a given sequence of probability measures µ n , and a sequence of real-valued random variables X n , and a constant c, -When h < h c , the second largest is power law: a -1 N M |τ | converges in distribution to a nontrivial random variable.

-When h = h c ∈ (h , h ), the largest jump has a polynomial size: a -1 N M |τ | is tight.

-When h > h c , the largest jump has logarithmic size: there exists c > 0 such that

M |τ | c log N ξ Ψ,h N -----→ N →+∞ 1.
(4.2.17) * See the list of symbols and abbreviations.

These two propositions described the big jump/localization phase transition. In a nutshell, when h < h c , there is a unique "big jump", with linear size (the other jumps are o(N )), whereas when h > h c , all jumps have at most logarithmic size. We finish with a fluctuation result. Fluctuations of the largest jumpM |τ | can also be obtained. They derive from the next subsection's results, dealing with the asymptotic behaviour of the largest jump under the constrained measure µ N,L , together with Proposition 4.5.2, which provides an approximation of ξ Ψ,h N as a mixing of the measures µ N,L in a rather explicit and more usable fashion than the original (4. 1.14). We refer to 4.5.4 for some details. Fluctuations or more precise path properties under ξ Ψ,h N when h ∈ (h , h ) depend on the behaviour of H and Ψ on the boundary of I . We do not investigate it deeper since the phases h ∈ (h , h ) are actually excluded in the three physical models we consider, but the proofs below can easily be adjusted.

Largest Jump for the constrained PS model

In the following paragraphs, we investigate the size M L of the largest jump for the constrained PS model µ N,L , that is the PS model with a fixed number of contact points. Under the measure P, that is for L independent jumps with distribution K, the size of the largest jump is power law, roughly N 1/α (Proposition 4.A.6). How does the conditioning τ L = N affects it? Of course, the behaviour depends on the value of the asymptotic density = lim L N : whether it is smaller or larger than c . The question has been deeply studied by Armendariz et al. (see for instance [4,[START_REF] Armendáriz | Thermodynamic limit for the invariant measures in supercritical zero range processes[END_REF]) from a slightly different point of view: the measure µ N,L is also the equilibrium measure of a particular zero-range process. In this setting, the big jump transition corresponds to a condensation phenomenon. In a word: Theorem 4.2.15. For any ∈ [0, 1],

M L N µ N,L -----→ N →+∞ L/N → 1 - c if < c , 0 if > c . (4.2.19)
The second largest jump has size o(N ) in probability.

The result is not fully satisfactory, particularly for > c . In fact, the investigation conducted in [4,[START_REF] Armendáriz | Thermodynamic limit for the invariant measures in supercritical zero range processes[END_REF] goes way deeper, providing refined limit theorems and fluctuations results for M |τ | . The critical regime = c is the most subtle and it requires a more difficult investigation.

Subcritical and supercritical density

We start with the subcritical regime ( < c ), for which the event τ L = N corresponds to non Cramer deviations for the random walk (τ n ) and produces a condensation phenomenon (or big jump phenomenon, depending on the perspective), that is, the first line in (4.2.19). Proposition 4.2.16 (Subcritical density regime, [START_REF] Armendáriz | Thermodynamic limit for the invariant measures in supercritical zero range processes[END_REF]). Suppose E[η] < +∞ and L N → ∈ [0, c ). Then the largest jump has linear size

M L N -LE[η] µ N,L ---→ 1. (4.2.20)
And, if L → +∞, its fluctuations satisfy

M L -(N -LE[η]) b L µ N,L = == ⇒ -X α . ( 4

.2.21)

As for the second largest jump, for any t 0, we have

µ N,L M L a L t ∼ P M L a L t -→ L→+∞ exp - 1 αt α . (4.2.22)
Remark 4.2.17. Note that convergence (4.2.20) still holds when L stays bounded.

Contrariwise, the supercritical regime corresponds to Cramer deviations of (τ n ): the atypical event τ L = N is typically accomplished when all the jumps are small (actually the event is typical for jumps with a distribution tilted by an exponential decay). Then the largest jump has logarithmic size instead of power law: Proposition 4.2.18 (Supercritical density regime). Suppose α 0 and L N → ∈ ( c , 1). Consider the unique x > 0 such that E x [η] = . There exists a sequence of integers (z L ), with z L ∼ log L

x , and a bounded sequence s L c > 0 (these sequences also depend on N ), such that, for any k ∈ Z,

µ N,L (M L -z L k) ∼ exp(-s L e -xk
µ N,L (M L -z L = k) ∼ exp(-s L e -xk ) -exp(-s L e -x(k-1) ). (4.2.24)
The sequences z L , s L are explicit (see the proof). The sequence (s L ) does not necessarily converge: if not then neither does µ N,L (M Lz L k). Actually, considering the unique

x L > 0 such that E x L [η] = N L (note that x L → x), one has µ N,L (M L -z L k) ∼ P x L (M L -z L k). (4.2.25)
In other words, once the distribution of the jumps is tilted, the conditioning does not affect the largest jump.

Critical regime

This paragraph is devoted to the critical regime, that is

N L -→ E[η]. (4.2.26)
Some of the following results are from [4]. The critical regime is the most demanding, and different behaviours emerge, depending on the parameter α. We will need to distinguish upside moderate deviations (when N LE[η]) and downside moderate deviations (when

N LE[η]). Proposition 4.2.20 (Critical regime, α ∈ (1, 2)). Write N = LE[η] + u L a L . 1. If u L → u ∈ R then for any w > 0, µ N,L M L a L w -----→ L→+∞ e -1 αw α g α,w u + 1 (α-1)w α-1 g α (u) . (4.2.27)
where g α is the density of X α and, for w > 0, g α,w the density of the random variable X α,w whose characteristic function is We expect it to also be the right answer for µ N,L , but P x L (τ L = N ) and Px L (τ L = N ) are not under control.

E[e itXα,
Note that when α ∈ (1, 2), the size if the largest jump M L and the deviations of τ L are of the same order under P. That's why the conditioning τ L = N affects the size if the largest jump, even for moderate deviations. It is not the case when α > 2, for the largest jump size is much smaller the typical deviations of τ L . The two following results are from [4], who only investigates the regime α > 2. The first one concerns upside deviations whereas the second one, which can be slightly extended, concerns downside deviations. We provide a proof for the second one, for which one case was not entirely addressed in [4].

N = LE[η] + σ (α -2)L log L 1 + α + 1 2(α -2) log log L log L - log ϕ( √ L log L) (α -2) log L + u L log L , ( 4
. If u L → -∞, then µ N,L M L a L t ∼ P M L a L t -----→ L→+∞ exp - 1 αt α .
(4.2.31)

3. If u L → u ∈ R then there exists p u ∈ (0, 1) such that

M L N -LE[η] µ N,L = ==== ⇒ L→+∞ Ber(p u ), (4.2.32)
with p u → 0 when u → -∞ and p u → 1 when u → +∞.

Proposition 4.2.23 (Critical regime, η ∈ L 2 : downside deviations). Suppose α 2 and

σ 2 = Var(η) < +∞. Write N = LE[η] -θ L , with 0 θ L L. 1. If θ L L a L , then µ N,L M L a L t ∼ P M L a L t -----→ L→+∞ exp - 1 αt α . (4.2.33) 2. If a L θ L σ 2 L → ω ∈ (0, +∞), then µ N,L M L a L t -----→ L→+∞ exp - +∞ t e -ωs s -1-α ds . (4.2.34) 3. If θ L L a L , there exists a sequence A L → +∞ and y L = θ L σ 2 L → 0 such that µ N,L (y L (M L -A L ) t) -----→ L→+∞ e -e -
N = LE[η] + √ LK L 1 + u L K 2 L , (4.2.36)
for a non-explicit sequence 1 K L √ log L which depends on the slowly varying function ϕ. When α ∈ {1, 2} with Var(η) = +∞, a more subtle analysis should be conducted, to obtain an analogous of Proposition 4.2.20 with a presumably different scaling. However we leave these predictions as conjectures.

Discussion: phases and phase transitions in the kernel models

For the constrained PS model, as well as for the convoluted models, the phase transition between the big jump phase and the localized phase has been fully described, and the free energy critical exponent computed in the previous subsections. We mentioned in the introduction other types of phase transition for the kernel models. They can occur if h or h is finite. We will not go through much details since these do not happen in the physical models. Though let us sketch what phenomena can emerge. Four types of phases can occur, corresponding to different values of the contact density (see Proposition 4.2.10).

(i) Delocalized phase (Figure 4.3): when = 0. The number of contact points is o(N ).

There is a unique macroscopic loop of size N . This occurs if and only if a = 0 and H (0) < +∞.

(ii) Big jump phase (Figure 4.4): when 0 < < c . The number of contact points is of order N . There is a unique big loop of size c( )N , the size of the other loops is power law.

(iii) Localized phase (Figure 4.2): when > c . The number of contact points is of order N , as well as the total loop length, but the loops are typically small (logarithmic size).

(iv) Density saturation: when = a ∈ (0, 1) (resp. = b ∈ (0, 1)). It is a phase where the density of contact points stays constant. It can happen in the localized phase (if a > c ) or in the jump phase phase (when a < c ), and in this case the normalized size of the big jump (1-a/ c ) is also saturated. This phase exists when H (a) < +∞ (resp. H (b) > -∞). Depending on Ψ, and particularly on the smoothness of H, different phase transitions between the phases described above occur, and the corresponding critical exponent for f H can be computed in each case. Let us give one example when h > -∞ (for instance if H(x) = x(1x)). Define U (x) = g(x) + H(x) and suppose that when x a, one has U (x) = U (a)c(xa) λ (1 + o(1)) for some c, λ > 0 (typically λ = 1). Then, for h h ,

f H (h) = U (a) + ah. ( 4

.2.37)

And for h h , The corresponding phase transition for the trajectories depends on a and α.

f H (h) = U (a) + ah + C(h -h ) λ+1 λ 1 h h (1 + o(1)). ( 4
-When a > 0, it is a contact density saturation.

-When a = 0 and E[η] < +∞, it is a the delocalization/big jump phase transition.

-When a = 0 and α < 1, then h c = h ∈ R and it is a delocalization/localization phase transition. Writing H (x) = H (0)-cx θ (1+o(1)) as x 0, one has λ = min(θ, κ-1)

and the critical exponent for f H is min(1 + 1/θ, 1/α). -We leave for now the convergence (4.2.1), and we define g by the infimum in (4.2.1). For any ∈ (0, 1), the function g (x) = x + log E[e -xη ] is strictly convex on R + . Furthermore, with E[e -ηx ] ∼ P(η = 1)e -x , we get lim x→+∞ g (x) = +∞. Therefore its minimum is reached at a unique point x ∈ R + . We compute the derivative:

Proofs: free energy

g (x) = 1 - E[ηe -xη ] E[e -xη ] = 1 -E x [η]. (4.3.1) If c = 1/E[η]
then g is nondecreasing and g( ) = g (0) = 0. If > c then the minimum is reached at a point x > 0, which is the only zero of g on (0, +∞). Since ( , x) → g (x) is is C 1 on R × R + , and analytic on R × R * + , with the implicit function theorem, → x is continuous, and analytic on ( c , 1), and so is g. Moreover, → x is increasing on ( c , 1), and x → +∞ when → 1. Then g is decreasing on ( c , 1), and g ( ) = log E[e -x η ] → -∞ when → 1.

-We finish with the convergence (4.2.1). When < c , it is a consequence of (4.A.46), whereas for > c , it can be deduced from Lemma 4.4.3 (see also [START_REF] Borovkov | Probabilities of large deviations of sums of independent random vectors on the boundary and outside of the Cramér zone. I[END_REF]). For = c , it is slightly more technical and we admit for now. If c > 0, fix > 0 sufficiently small. Then let

L 1 , L 2 , N 1 , N 2 such that L 1 + L 2 = L, N 1 + N 2 = N , L 1 N 1 → (1 -), L 2 N 2 → (1 + ), L i L → 1 2 , and N i N → 1 2 . Observe that P(τ L = N ) P(τ L 1 = N 1 )P(τ L 2 = N 2 ). (4.3.2) Thus lim inf P(τ L = N ) 1 2 g( c + ), (4.3.3)
which vanishes when 0. Thus we get the convergence (4.2.1). On the other hand, when c = 0, and L N → 0, just write log P(τ L = N ) (L -1) log K(1) + log K(N -L -1).

We add some useful properties to Lemma 4.2.1. -If α > 1 and Var(η) = +∞, then g is C 2 on [0, 1), with g ( c ) = 0.

-

If = c with σ 2 = Var(η) < +∞, then lim c g ( ) = -( 3 c σ 2 ) -1 . -If > c then let x = x be such that E x [η] = 1 , and σ 2 x = Var x (η). Then g ( ) = -( 3 σ 2 x ) -1 .
Proof. We know that g is analytic on [0, 1)\{ c }. We compute g ( ) for > c . When > c , since x solves E x [η] = 1/ , one gets

∂ x = - 1 2 1 (∂ x E x [η]) |x = -1 2 (-Var x (η)) = 1 2 σ 2 x . (4.3.4) Hence, g ( ) = ∂ x × (∂ x log E[e -xη ]) |x = 1 2 σ 2 x (-E x [η]) = -1 3 σ 2 x . (4.3.5)
And this goes to -1 3 σ 2 as c .

Instead of Proposition 4.2.3, we prove the following stronger result.

Lemma 4.3.2. The following hold as c , for a certain slowly varying function φ (depending on α and ϕ).

1. If α ∈ [0, 1), g( ) ∼ - 1 1-α φ (1/ ) , g ( ) ∼ - 1 1 -α α 1-α φ (1/ ) . (4.3.6) 2. If α = 1, lim c log g( ) log( -c ) = +∞, lim c log g ( ) log( -c ) = +∞. (4.3.7) 3. If α ∈ (1, 2] and σ 2 = +∞, g( ) ∼ -( -c ) α α-1 φ ( -c ) -1 , g ( ) ∼ - α α -1 ( -c ) 1 α-1 φ ( -c ) -1 . (4.3.8) And, if α = 2, φ(u) → 0 when u → +∞. 4. If α 2 and σ 2 < +∞ then g( ) ∼ -C( -c ) 2 , g ( ) ∼ -2C( -c ). (4.3.9)
Proof. We know that

g( ) = g (x ) = x + log E[e -x η ], g ( ) = ∂ g (x ) = log E[e -x η ], (4.3.10)
where, when > c , x is the unique positive solution of

E[ηe -x η ] E[e -x η ] = 1 , (4.3.11)
which vanishes as c . We write = c + δ and use the estimates of Lemma 4.A.5 to obtain a equivalent of x . The asymptotic behaviour of g will immediately follow thanks to

g ( ) ∼ E[e -x η ] -1. (4.3.12) First note that when c > 0, E[η] -1 ∼ E[η] 2 δ. 1. If Var(η) < +∞ then E[ηe -xη ] E[e -xη ] = E[η] -E[η 2 ]x(1 + o(1)) 1 -xE[η](1 + o(1)) = E[η] -x Var(η)(1 + o(1)). (4.3.13) Thus x ∼ E[η] 2
Var(η) δ. We find the desired expansion by injecting it in the expression of g . Proof. Both H (Assumptions 4.1.4) and g (Lemma 4.2.1) are concave on [0, 1], and H is strictly concave. Therefore, so is U . As a strictly concave function, → h + U ( ) has a unique maximum point (h) ∈ I . Since its left derivative at = 1 is -∞ (Lemma 4.2.1), the minimum is actually reached in I \{1}.

-When h h , the function

→ h + U ( ) is decreasing on I \{1}. So (h) = a and f H (h) = ha + U (a): it is analytic on (-∞, h ).
-Similarly, when h

h , → h + U ( ) is increasing on I \{1}, so (h) = b and f H (h) = hb + U (b). -When h ∈ (h , h ), (h) ∈ (a, b): it solves U ( (h)) = -h. Since U is nonincreasing, h → (h) is nondecreasing. -The function (h, ) → h + U ( ) is continuously differentiable on R × (a,
(h) = c , that is, since (h) solves U ( (h)) = -h, as long as h = h c := -U ( c ).
Proof of Proposition 4.2.8. Recall the definitions

f H (h) = sup ∈[0,1] { h + H( ) + g( )}, (f H ) reg (h) = sup ∈[0,1] { h + H( )}. ( 4
( ) = H( c ) + H ( c ) -A 2 + o( 2 ), (4.3.26)
with A, B > 0 by concavity, and H ( c ) = -h c by definition. Hence the function U admits the following expansion:

U ( ) = H( c ) -h c -A 2 -B 2 1 0 + o( 2 ). (4.3.27)
Similarly, for the derivative,

U ( ) = -h c -2A -2B 1 0 + o( ). (4.3.28) It yields h + U ( ) = (h -h c ) -2A -2B 1 0 + o( ). (4.3.29)
Recall that (h) solves h + U ( ) = 0. Therefore, when h is close to h c , the difference ˆ = ˆ (h) := (h)c has the sign of g := hh c . More precisely, as g → 0,

ˆ = g 2A + 2B1 g 0 + o(1) = g 2A + 2B1 g 0 + o(g). (4.3.30)
Rewrite the free energy as

f H (h) = h (h) + U ( (h)) = h (h) + H( c ) -h c ˆ -(A + B1 ˆ 0 )ˆ 2 + o(ˆ 2 ) = (h c + g)( c + ˆ ) + H( c ) -h c ˆ (h) -(A + B1 ˆ 0 )ˆ 2 + o(ˆ 2 ) = h c c + H( c ) + c g + gˆ -(A + B1 ˆ 0 )ˆ 2 + o(ˆ 2 ). (4.3.31)
We also know that

h c c + H( c ) = f H ( c ) Then, with the expansion (4.3.30) of ˆ (h), f H (h) =f H ( c ) + c g + 1 2 g 2 1 2A + 2B1 g 0 + o(g 2 ) =f H ( c ) + c g + 1 4A g 2 - B 4A(A + B) g 2 1 g 0 + o(g 2 ). ( 4
(η) = ∞. Recalling Lemma 4.3.2, U ( ) = H( c ) -h c -A 2 -2 φ(1/ )1 0 (1 + o(1)) + O( 3 ), (4.3.33)
U ( ) = -h c -2A -2 φ(1/ )1 0 (1 + o(1)) + O( 2 ), (4.3.34)
where φ slowly varies and φ → 0. We know that, when g 0,

g = h -h c = 2Aˆ + 2ˆ φ(1/ˆ )(1 + o(1)). ( 4
(h) = h c c + H( c ) + c g + gˆ -Aˆ 2 -ˆ 2 φ(1/ˆ )(1 + o(1)) = f H ( c ) + c g + 1 4A g 2 - 1 4A 2 g 2 φ(1/g)(1 + o(1)). (4.3.37)
And similarly, when h → h c ,

(f H ) reg (h) = f H ( c ) + c g + 1 4A g 2 + O(g 3 ). (4.3.38) Eventually, recall (Lemma 4.3.2) g( ) ∼ -( -c ) α α-1 φ ( -c ) -1 , (4.3.39)
(here κ = 2 (4.2.3)). Therefore,

f H (h) -(f H ) reg (h) ∼ - 1 4A 2 (h -h c ) 2 φ(1/(h -h c )) ∼ c g c h h c . (4.3.40) with c = (h c /(2A c )) 2 .
with y L = max x i = x i 0 (with i 0 minimal if the maximum is not unique) and y i 0 = x L . In this way the biggest coordinate of T ((

x i ) L i=1 ) is the last one. Let C ⊂ {η L > max j L-1 η j }. By symmetry, µ N,L • T -1 (C) = Lµ N,L (C). (4.4.7)
The same goes for π N,L :

π N,L • T -1 (C) = Lπ N,L (C). (4.4.8)
Now, let t N be such that b N t N < c N /2 and t N N . Introduce the set

D N,L = {r ∈ N : |N -LE[η] -r| t N } , (4.4.9)
and consider the event

B N,L = η L ∈ D N,L ; max 1 j L-1 η j < t N ⊂ {η L > max j L-1 η j }. ( 4

.4.10)

Let A ∈ B(R L-1 ). To simplify, we will also write A for the event {(η j ) L-1 j=1 ∈ A}. One has

µ N,L • T -1 (A ∩ B N,L ) = Lµ N,L (A ∩ B N,L ) = L P (A ∩ B N,L ∩ {τ L = N }) P (τ L = N ) . ( 4

.4.11)

By assumption, we know that

sup L V N P(τ L = N ) LK(N -LE[η]) -1 -→ N →+∞ 0. (4.4.12)
As a consequence, as N → +∞, sup

1 L V N sup A∈B(R L-1 ) µ N,L • T -1 (A ∩ B N,L ) - P (A ∩ B N,L ∩ {τ L = N }) K(N -LE[η]) -→ N →+∞ 0.
(4.4.13) We compute the probability

P (A ∩ B N,L ∩ {τ L = N }) = r∈D N,L K(r)P   A ∩    L-1 j=1 η j = N -r    ∩ max 1 j L-1 η j < t N   . ( 4
L V N sup r∈D N,L K(r) K (N -LE[η]) -1 -→ N →+∞ 0. (4.4.15)
Hence, as N → +∞, and uniformly in L c (Nc N ) and A ∈ B(R L-1 ),

P (A ∩ B N,L ∩ {τ L = N }) K(N -LE[η]) ∼ r∈D N,L P   A ∩    L-1 j=1 η j = N -r    ∩ max 1 j L-1 η j < t N   = P   A ∩    L-1 j=1 η j -LE[η] t N    ∩ max 1 j L-1 η j < t N   . (4.4.16)
The last two events (between braces) occur with high probability as N → +∞. It is a consequence of Proposition 4.A.7 (recall t N b N ):

inf L c(N -c N ) P   L-1 j=1 η j -LE[η] t N   -→ N →+∞ 1; (4.4.17)
and Proposition 4.A.6:

inf L c(N -c N ) P max 1 j L-1 η j < t N -→ N →+∞ 1. (4.4.18)
Eventually,

sup 1 L c(N -c N ) sup A∈B(R L-1 ) µ N,L • T -1 (A ∩ B N,L ) -P(A) -→ N →+∞ 0. (4.4.19) By taking A = R L-1 we get sup 1 L c(N -c N ) µ N,L • T -1 (B N,L ) -1 -→ N →+∞ 0. (4.4.20)
Also with Corollary 4.A.7,

sup 1 L c(N -c N ) π N,L • T -1 (B N,L ) -1 -→ N →+∞ 0. (4.4.21)
Eventually, with the same uniformity as in the previous lines,

µ N,L • T -1 (A ∩ B N,L ) µ N,L • T -1 (A) π N,L • T -1 (A ∩ B N,L ) π N,L • T -1 (A). (4.4.22)
The statement of the proposition is a consequence of the symmetry of the L coordinates (η j ) (see (4.4.7)).

As a corollary we get Proposition 4.2.16, Proposition 4.2.20 item 2, and Proposition 4.2.22 item 1. We prove it briefly.

Proof. The proof is the same for these three cases. The only point to observe is that in both of them, N -LE[η] is, by assumption, much larger than a L and b L . Denote by πN,L the distribution, under P, of

  η 1 , . . . , η L-1 , N - L-1 j=1 η j   , ( 4
A = ηL -(N -LE[η]) b L t = - L-1 j=1 η j -LE[η] b L t . ( 4

.4.24)

And, for any c L → +∞, such that LP(η c L ) is bounded,

sup k∈Z σ 2 x L Px L (τ L = k) -f N k -LE x L [η] σ 2 x L -→ 0, (4.4.32)
where f N is the standard Gaussian density function.

Proof. Thanks to [29, Th. 1.2], it is sufficient to prove a central limit theorem for P x L and Px L . Let us prove it briefly for Px L (the other case is recovered by taking c L = ∞). With dominated convergence, since c L → +∞, and x L → x, we obtain

Êx L [η] → E x [η] and Êx L [η 2 ] → E x [η 2 ]. (4.4.33) Therefore, Êx L (η -E x L [η]) 2 -→ σ 2 x . (4.4.34)
On the other hand, with Schwarz inequality, one has

√ L( Êx L [η] -E x L [η]) = √ LE x L [η1 η>c L ] [LP x L (η > c L )]E x L [η 2 1 η>c L ] 1/2 -→ L→+∞ 0, (4.4
.35) because the first factor is bounded, by assumption; and the second, by dominated convergence, vanishes as L → +∞. We then compute the characteristic function, which, thanks to (4.4.34), can be developed

Êx L exp it η -E x L [η] σ 2 x L = 1 + it σ 2 x L ( Êx L [η] -E x L [η]) - t 2 2σ 2 x L Êx L (η -E x L [η]) 2 (1 + o(1)).
(4.4.36) And combining these three lines, we get

Êx L exp it η -E x L [η] σ 2 x L L -→ exp(-t 2 /2). ( 4
[η] = 1 . We compute LP x L (η > k) = L E[e -x L η ] +∞ n=k+1 ϕ(n) e -x L n n 1+α ∼ L→∞ L E[e -xη ] +∞ n=k+1 ϕ(n) e -x L n n 1+α . ( 4

.4.38)

Introduce

A L = 1 x L log L -(1 + α) log log L + (1 + α) log x + log ϕ(log L) -log E[e -xη ] . (4.4.39)
In particular A L → +∞ and it is chosen to satisfy

Lϕ(A L )e -x L A L A 1+α L E[e -xη ] -→ L→+∞ 1. (4.4.40)
So, for any k ∈ Z,

LP x L (η > A L + k) ∼ L→∞ L E[e -xη ] +∞ n=k+1 ϕ( A L + n) e -x L ( A L +n) ( A L + n) 1+α ∼ Lϕ(A L )e -x L A L A 1+α L E[e -xη ] +∞ n=k+1 ϕ( A L + n) ϕ(A L ) e -x L (n-{A L }) (1 + n A L ) 1+α ∼ +∞ n=k+1 e -x(n-{A L }) = e -x(k+1-{A L }) 1 -e -x .
(4.4.41)

Eventually, with Lemma 4.4.3,

Px L (τ L = N ) P x L (τ L = N ) -→ L→∞ 1. (4.4.42)
Combining these estimates, we get the result:

µ N,L (M L -A L k) ∼ e -e -x(k+1-{A L }) 1-e -x ∼ P x L (M L -A L = k), (4.4.43)
which is the statement of the proposition, with z L = A L and s L = (1e -x ) -1 e x({A L }-1) . Its continuous density will be denoted by g α,w . Uniformly in w in any compact subset of (0, +∞), and uniformly in k ∈ N,

a L P(τ L = k) -g α,w k -L Ê[η] a L -→ L→+∞ 0. (4.4.46)
Proof. For convenience we write c L = wa L . We compute the characteristic function

Ê[e it(τ L -L Ê[η]) ] = Ê[e it(η-Ê[η]) ] L . (4.4.47) One has Ê e it(η-Ê[η]) = wa L n=1 K(n)e it(n-Ê[η]) 1 P(η wa L ) = 1 + 1 P(η wa L ) wa L n=1 ϕ(n) n 1+α (e it(n-Ê[η]) -1 -it(n -Ê[η])). (4.4.48) Thus Ê e it a L (τ L -L Ê[η]) = 1 + 1 P(η wa L ) wa L n=1 ϕ(n) n 1+α e it n-Ê[η] a L -1 -it n -Ê[η] a L L ∼ 1 + ϕ(a L ) a α L 1 a L wa L n=1 ϕ(n) ϕ(a L ) n a L -(1+α) e it n-Ê[η] a L -1 -it n -Ê[η] a L L -→ exp w 0 e ity -1 -ity y 1+α dy =: v α,w (t).
(4.4.49) Therefore the convergence (4.4.44) is proved. One readily see that v α,w ∈ L 1 , so the limit X α,w has a continuous density g α,w . Then we need to prove the local limit theorem. Let

f (t) = E[e itη ], ψ(t) = E[e it(η-E[η]) ], (4.4.50)
and The same proof as for Proposition 4.A.8 works: write

f (t) = Ê[e itη ], ψ(t) = Ê[e it(η-Ê[η]) ], (4.4
2πa L P(τ L = k) = πa L -πa L e -izu ψ u a L L du, (4.4.53)
where

z = z k,L = k-L Ê[η] a L
. The only point that must be adapted is the control of

J = A |u| πa L e -izu ψ u a L L du. (4.4.54)
For t ∈ [-π, π], and L sufficiently large,

| ψ(t)| = | f (t)| = 1 P(η wa L ) +∞ n=1 e int K(n) - n>wa L e int K(n) 1 1 -Cw -α L |f (t)| + Cw -α L 1 1 -Cw -α L |f (t)| ∨ 1 2 1 + 2Cw -α L |f (t)| ∨ 1 2 1 + Cw -α L . ( 4
a L P(τ L = N ) = g α N -LE[η] a L + o(1) -→ L→+∞ g α (u). (4.4.60)
On the other hand, with Proposition 4.4.4,

a L P(τ L = N ) = g α,w N -L Ê[η] a L + o(1), (4.4.61) with N -L Ê[η] a L = u L + L E[η] -Ê[η] a L . (4.4.62)
We readily see that

lim L→+∞ L E[η] -Ê[η] a L = lim L→+∞ L a L E[η1 η>wa L ] = 1 (α -1)w α-1 . (4.4.63) So a L P(τ L = N ) -→ L→+∞ g α,w u + 1 (α -1)w α-1 . (4.4.64)
On the other hand, We want

P(η wa L ) L = 1 - ϕ(a L ) αw α a α L (1 + o(1)) L -→ L→+∞ e -1 αw α . (4.4.65) Eventually, µ N,L (M L wa L ) -→ L→+∞ e -1 αw α g α,w u + 1 (α-1)w α-1 g α (u) . ( 4
LP x L (η c L ) ∼ L n c L ϕ(n) e -nx L n 1+α (4.4.69)
to be of order one. Then the result will follow from Lemma 4.4.3.

1. If a L θ L σ 2 L → ω ∈ [0, +∞), then LP x L (η ta L ) ∼ Lϕ(a L ) a α L 1 a L n ta L ϕ(n) ϕ(a L ) e -n a L a L x L ( n a L ) 1+α -→ +∞ t e -ωs s -1-α ds, (4.4.70)
and we get the first two cases of the proposition.

If a

L θ L σ 2 L → +∞, then set β L = log L + α log x L + log ϕ 1 x L , (4.4.71)
B L = 1 x L β L -(1 + α) log β L + log ϕ( β L x L ) ϕ( 1 x L ) . ( 4

.4.72)

With the definition of a L (Lemma 4.A.4), ) . Then The first ratio converges (in probability) to a constant because |τ |/N → in probability and a L is regularly varying. And we conclude with Slutsky's theorem.

e β L = Lx α L ϕ(1/x L ) ∼ (a L x L ) α ϕ(1/x L ) ϕ(a L ) . ( 4
LP x L (x L (η -B L ) t) ∼ Le -x L B L ϕ(B L ) x L B α+1 L x L n B L +t/x L ϕ(n) ϕ(B L ) e -(n-B L )x L 1 + (n-B L )x L x L B L 1+α .
The same methods gives the logarithmic asymptotic behaviour of the largest jump when h > h c . Now we want to prove Proposition 4.2.13 about Gaussian fluctuations for the number of contact points. In fact, we prove a slightly stronger result. From the decomposition (4.1.14), we know that ξ Ψ,h N is a mixing of the measures µ N,L . Thanks to Lemma 4.5.1, we know that this mixing can be restricted to L = N ± N . In fact, when h ∈ (h , h ), it can be restricted to L = N + O( √ N ) and the next proposition makes it rather explicit. It basically states that ξ Ψ,h N µ N,L , where the number of contact point L is random, concentrated around N with approximately Gaussian deviations. where Y ∼ N (0, 1) is independent of the stable random variable X α (see Proposition 4.A.7 for the definitions b L and X α ).

4.A Appendix: Results for heavy tail random walks

4.A.1 Slowly Varying Functions and Laplace Transform Estimates

A complete reference for slowly varying functions is [START_REF] Bingham | Regular variation[END_REF]. We recall some of the results which are useful in this work.

4.A.2 Laplace Transform

We will need the precise behaviour of K Laplace transform at 0. The following lemma is a consequence of [14, § 1.5.6 and Cor. 1.7. -X α is an α-stable random variable, whose continuous and bounded density function is denoted by g α .

E[e itXα ] = exp |t| α e -i sgn(t)

4.A.4 Local Limit Theorems

We start with the most standard local limit theorem for discrete random variables, for "moderate deviations". The definitions of µ L , b L and g α are in Proposition 4.A.7. One can find a proof in [60, Ch. 9 § 49 and § 50], except for the case α = 2 and Var(η) = +∞, for which we refer to [START_REF] Doney | A bivariate local limit theorem[END_REF]. Note that it also holds for any sum of IID random variables with finite variance. For it is useful in the proof of Proposition 4.2.20, we give, following [START_REF] Gnedenko | Limit distributions for sums of independent random variables[END_REF], a complete proof of this result when α ∈ (1, 2).

Proof when α ∈ (1, 2). We compute the characteristic function We will need the following two lemmas, whose proofs are postponed after the proposition's proof. The local limit theorem is proved.

Proof of Lemma 4.A.9. We know the expression of v α (t): and the equation ∂ h f(0, h) = is solvable only for c > 0, with analytic dependence for > c , and the variational problem is solved by h = 0 for < c (and of course g(0, ) = 0 then). Therefore g(0, ) = 0 for ∈ [0, c ] and g(0, ) < 0 for > c (Figure 5.2). This transition is the big jump transition. Let us switch the disorder on (β > 0):

0 h f(h) 1 c = 0 g( )
-the smoothing bound (3.2.10) implies continuity of ∂ h f(β, •). Hence the singularity arising from the mechanism in H2 is no longer present and the optimization procedure becomes the one in H1, even when E[τ 1 ] is finite. That is, the equation ∂ h f(β, h) = has a unique solution for every ∈ (0, 1): there is no switch (unlike in H2) between a regime in which ∂ h f(β, h) = is solvable and one in which it is not solvable, like for the big jump transition (Figure 5.3).

-However, we cannot conclude, like in H1, that the free energy is analytical. The main obstruction is that we do not know whether f(β, •) is analytic in the localized regime.

Nevertheless, the arguments we have just outlined strongly suggest that the big jump transition is suppressed by the disorder. And we will actually show that this is the case by analyzing the trajectories of the disordered system, in Section 5.2.2.

On a more subordinate note, the behaviour of g(β, ) when 0 can be deduced from f(β, h) critical behaviour (see Proposition 5.3.5). In a nutshell, g(β, ) should look like Figure 5.3. Once the behaviour of g(β, ) is (partially) understood, it can be injected into the variational formula (5.2.5) to obtain properties of f H (β, h).

Let us make more precise this informal discussion: (5.2.8)

3. Moreover the map → g(β, ) is analytic on (0, 1) if and only if h → f(β, h) is analytic on (h c (β), +∞). In this case, also f H (β, •) is analytic on R.

The step from smoothness to analyticity in disordered systems is obstructed by the so called Griffiths singularities: we refer to [START_REF]The localized phase of disordered copolymers with adsorption[END_REF] for a discussion and for relevant literature on this issue which, however, is essentially fully open for the PS model.

Path Properties

Like in the homogeneous models, the contact density satisfies a law of large number. (5.2.10)

From our perspective, the largest jump plays a crucial role:

M |τ | = max 1 k |τ | (τ k -τ k-1
).

(5.2.11)

We are in fact going to show that when β > 0, unlike for the homogeneous models and whatever the value of α is, all the jumps are small. where disorder irrelevance is predicted (at least for α ∈ (1, 2)) based on a direct application of Harris criterion (see Section 3.2) that we can easily reproduce here. Their argument relies on the hyperscaling relation (see [52, § 5.3.1]), which in our context can be written νd = κ, where κ is the critical exponent for the free energy f H (h) (see Proposition 4.2.8 and definition (4.2.3)). Therefore, Harris criterion states that disorder is irrelevant for α ∈ (1, 2), and marginal for α 2. It would be extremely interesting to be able to adapt or to make sense of the Harris argument for fixed contact number and Kernel models, even just at a heuristic level.

Remark 5.2.5. For completeness we recall that for the disordered PS in the localized phase, all jumps have finite length (more precisely, the largest jump is of length O(log N ): see [START_REF]The localized phase of disordered copolymers with adsorption[END_REF]): in particular we have We mention that the delocalized phase * there are only very few contacts and close to the boundary [START_REF]Path properties of the disordered pinning model in the delocalized regime[END_REF][START_REF] Giacomin | Estimates on path delocalization for copolymers at selective interfaces[END_REF] and

M |τ | N ν h,ω,β N -----→ N →+∞
1 in P probability for h < h c (β).

(5.2.15)

But this is a delocalization transition and it has nothing to do with a big jump transition.

We complete this section by signaling that in the zero range context the effect of disorder has been considered in [START_REF] Del Molino | Condensation in randomly perturbed zero-range processes[END_REF][START_REF] Grosskinsky | Instability of condensation in the zero-range process with random interaction[END_REF][START_REF] Mailler | Condensation and symmetry-breaking in the zerorange process with weak site disorder[END_REF]: the rates are chosen in such a way that the invariant measure of the process is, like in the non disordered case, given by a product measure. The particle conservation law of the dynamics imposes a global constraint like in the fixed contact number case we consider. The disorder is however of a rather different nature as it is indexed by the excursions: in our language, the disorder is introduced as a modification of the renewal probability K(•) that become themselves random. We refer to the original publications for the details, but we stress that also for some of the models in [START_REF] Del Molino | Condensation in randomly perturbed zero-range processes[END_REF][START_REF] Grosskinsky | Instability of condensation in the zero-range process with random interaction[END_REF][START_REF] Mailler | Condensation and symmetry-breaking in the zerorange process with weak site disorder[END_REF] the suppression of the big jump phenomenon due to the disorder has been established.

1

 1 tout δ > 0, µ n (|X n -c| δ) -----→ Produits de matrices aléatoires et Modèles d'Ising désordonnés Contents 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Généralités sur l'exposant de Lyapunov . . . . . . . . . . . . . . . . . . 1.2.1 Expression implicite et mesures invariantes . . . . . . . . . . . 1.2.2 Continuité de l'exposant de Lyapunov . . . . . . . . . . . . . . 1.2.3 Dérivabilité et Analyticité . . . . . . . . . . . . . . . . . . . . . 1.3 Le Modèle de Derrida-Hilhorst . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Développement « régulier » en puissances de 2 . . . . . . . . . 1.3.2 Singularité 2α . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Modèles en désordre faible . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Perturbation faible de la matrice I 2 . . . . . . . . . . . . . . . . 1.4.2 Limite continue du modèle de Derrida-Hilhorst : un modèle exactement soluble . . . . . . . . . . . . . . . . . . . . . . . . . 1.A Annexe : Démonstration des Théorèmes 1.4.1 et 1.4.3 . . . . . . . . . . 1.A.1 Aperçu de la méthode . . . . . . . . . . . . . . . . . . . . . . . 1.A.2 Trois lemmes sur l'opérateur M 2 et démonstration des théorèmes 1.A.3 Étude de l'opérateur différentiel M 2 et preuves des lemmes . . 1.A.4 Questions ouvertes et Commentaires . . . . . . . . . . . . . . .

σ 1 N

 1 N +1 ∈{-1,1} 1 σ N +1 =σ 1 N k=1 k (σ k , σ k+1 ). (1.1.8) C'est-à-dire Z N,β = Tr(L 1 • • • L N ) = e βN J+β N k=1 h k Tr(M 1, • • • M N, ).(1.1.9)Par suite, l'énergie libre par spin estf(β) := lim N →+∞ 1 N log Z N,β (ω) = β(J + E[h 1 ]) + lim N →+∞ log Tr(M 1, • • • M N, ). (1.1.10)

  Derrida-Hilhorst et ce sera notre objet d'étude principal (voir les sections suivantes). Leur étude se concentre sur le comportement du système lorsque les interactions deviennent très fortes (c.à-d. J → +∞ ou de façon équivalent → 0), toutes choses égales par ailleurs. Ils se placent dans le régime où E[h] > 0 (c.-à-d. E[log Z] < 0) et prédisent le phénomène suivant. Dans la limite J → +∞, les spins s'alignent majoritairement avec le champ et la proportions de spins -1 tend vers 0. En revanche la structure des configurations diffère selon la loi précise du champ et deux régimes principaux sont à distinguer. Lorsque E[e -2βh ] < 1 (c.-à-d. E[Z] < 1), la longueur moyenne des segments de spins -1 dans la chaine reste bornée. En revanche, dans le cas, statistiquement plus frustré, où E[e -2βh ] > 1 (c.-à-d. E[Z] > 1)

Figure 1 . 1 : 1 a a 2 +b 2 a a 2 +b 2 λ λ a 2 +b 2 ,-z 2 1 |1 + z 1

 111211 Figure 1.1: Les coefficients de couplage du modèle de McCoy et Wu. Cette définition assure l'invariance de chacune des réalisations par translation verticale, et leur invariance en loi par translation horizontale. Un calcul similaire à celui développé par Onsager fournit l'expression explicite de l'énergie libre par spin. Introduisons, pour θ ∈ R, les matrices aléatoires M β,n (θ) := 1 a a 2 +b 2 a a 2 +b 2 λ λ a 2 +b 2 , (1.1.13)

Figure 1 . 2 :

 12 Figure 1.2: Définition du paramètre α.

3 . 2 .

 32 La richesse de ce comportement est préservée par le passage à la limite. D'autre part, cette étude permet une meilleure compréhension des travaux de McCoy et Wu et rend rigoureuse une partie de leur approche ([26, Appendix]).

1 .

 1 A Annexe : Démonstration des Théorèmes 1.4.1 et 1.4.3

  .A.15) D'autre part, puisque ν η est une mesure invariante, et en utilisant le Lemme 1.A.1,

A. 55 )

 55 puis que ρ 0 admet ainsi un prolongement C ∞ en 0, toujours strictement positif, et satisfaisant l'équation (1.A.49). On ajuste enfin la constante α de sorte que ρ 0 d = 1.Preuve lorsque V 1 a deux directions stables. De nouveau on ne démontre que le Lemme 1.A.3. On suppose u 1 = u 2 = u 3 = 0, et E[u2 4 ] = 0. Alors la fonction b 1 a un zéro d'ordre deux en 0 et en π/2, et est strictement positive ailleurs. L'irréductibilité assure que δ 2 et δ 3 ne sont pas nuls. Disons, quitte à conjuguer, que δ 3 > 0. Les deux cas δ 2 > 0 et δ 2 < 0 doivent être traités séparément.Démonstration du Lemme 1.A.3 lorsque δ 2 > 0. On souhaite toujours résoudre

Proposition 2 . 3 . 1 .

 231 Pick an integer K ∈ A ∪ {0}, and fix β ∈ [K, K + 1]. The following expansion holds when goes to 0,

Lemma 2 . 5 . 5 . 1 BB- 1 +

 25511 Fix γ 1, C > 0, B > 1 and N ∈ N and set τ = γ B-C . One has

.5. 24 )

 24 Remark 2.5.8. If Z is not bounded but E[Z κ ] < +∞ for some κ > α then, with another choice of B and N , one can get the slightly weaker lower bound E[X α ] c log(1/ ) log log(1/ ) .

.5. 27 ) 2 . 5 . 3

 27253 Proof of Theorem 2.1.6

  (a) The random matrix M has non-negative entries. And, almost surely, there exists N 1 such that the product M N, • • • M 1, has positive entries. (b) There exists δ > 0 such that E[ N δ ] < 1 and E[ C δ ] < +∞. (c) E[log + L ] < +∞.

A. 27 )

 27 as long as (2.A.15) holds with β = K + 1.

Chapter 3 71 3. 3

 3713 An introduction to homogeneous and disordered Poland-Scheraga modelContents 3.1 The Homogeneous Poland-Scheraga Model . . . . . . . . . . . . . . . . 66 3.2 The Disordered Poland-Scheraga Model . . . . . . . . . . . . . . . . . . Beyond the Poland-Scheraga model . . . . . . . . . . . . . . . . . . . . 74

Figure 3 . 1 :

 31 Figure 3.1:In the top figure, DNA is in its native state: all bases are paired together. In the bottom, a loop opens and DNA is denaturated at this spot. In practice -and above all approaching denaturation -due to thermal fluctuations, there are several loops even when the two strands are still together. We will talk about denaturation only when the two chains are completely detached, except at some bases at finite distance of the extremities (in fact, we will always impose that the extremities are attached, and the thermal fluctuations at the boundary lead to having some contacts of the two strands, essentially only at a finite distance from the extremities). By passing to the complementary, also the notion of native state that we will use will be somewhat generalized.

.1. 2 )

 2 with ϕ(n) a slowly varying function: Definition 3.1.2. A function ϕ : (0, +∞) → (0, +∞) is called slowly varying if it is measurable and if for any a > 0,

τ 5 τ 9 -τ 8 τ 14 -τ 13

 91413

Figure 3 . 2 :

 32 Figure 3.2:A schematic view[START_REF]Disorder and critical phenomena through basic probability models[END_REF] of the correspondence between DNA and renewal process: the paired bases in DNA are the contact points of the renewal, and the loops correspond to inter-arrival of length 2 or more. The coiling structure is forgotten.

.1. 10 )

 10 so f(h) is the free energy (density) of the model. The proof of (3.1.10) follows from the Renewal Theorem formula (3.1.1

Figure 3 . 3 :

 33 Figure 3.3: By differentiating the free energy with respect to h one readily sees(3.1.11) that for h < 0 the density of contact sites is zero, while for h > 0 such a density is positive. But thanks to (3.1.8) one can go much beyond just by exploiting estimates on the renewal function (see[START_REF] Caravenna | Sharp asymptotic behavior for wetting models in (1+ 1)-dimension[END_REF][START_REF] Giacomin | Random polymer models[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF]). In fact ν N,h (A) can be always expressed in terms of ratios of partition functions. What one can show is that, in the limit N → ∞, the point process converges (in more than one sense, see again[START_REF] Caravenna | Sharp asymptotic behavior for wetting models in (1+ 1)-dimension[END_REF][START_REF] Giacomin | Random polymer models[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF]) to the renewal process τ (h) defined by the jump probability K h (•) (see the list after (3.1.6)). For h > 0, the Renewal Theorem formula (3.1.1) is sufficient to establish such a result, but sharp estimates on how the real function vanishes at ∞ are needed to recover such a result in the h < 0 case. For h > 0 (figure on the left), all loops are essentially of finite size (more precisely: the largest has size O(log(N )) and this logarithmic behaviour just reflects the exponentially decreasing behaviour of K h (n) for n → ∞). On the other hand, if h < 0 (figure on the right) the renewal points are finitely many and localized near the extremities: one huge loop, of size that differs from N only by a finite (random) number, occupies the whole system. At criticality, h = 0, all depends on K(•): we invite the interested reader to look at[START_REF] Caravenna | Sharp asymptotic behavior for wetting models in (1+ 1)-dimension[END_REF][START_REF] Giacomin | Random polymer models[END_REF][START_REF]Disorder and critical phenomena through basic probability models[END_REF].

Figure 3 . 5 :

 35 Figure 3.5: Free energy h → f(β, h) for β > 0.

Figure 3 . 6 :

 36 Figure 3.6: Schematic view of the localized phase of GPS model when the two strands fully bind together (item 1). On the right a zoom on the local structure: the jumps are O(log N ) and, of course, the loops must me asymmetric to allow such a localized behaviour despite the fact that the two strands have different lengths M > N .

Figure 3 . 7 :

 37 Figure 3.7: Schematic view of the big jump regime in the GPS model (item 2): a unique large loop emerges, the other loops are essentially small (see the zoomed figure on the right).

  (i) A bond segment of length k forms with probability proportional to e βke b = ω k b .

.1. 23 )

 23 Last but not least, the α-stable random variable X α and the sequences a L and b L (which also depend on α) are defined in the appendix (Lemma 4.A.4 and Proposition 4.A.7).About the kernel Ψ. The assumptions are a bit technical. In a word, we want Ψ(m, N ) exp(N H(m/N )) for a certain concave function H. Assumptions 4.1.4 (on the kernel Ψ). The function Ψ(m, N ) can be writtenΨ(m, N ) = exp N H m N Q(m, N ) 0,(4.1.24)where H and Q satisfy the following properties. (i) (a) The function H : [0, 1] → R ∪ {-∞} is strictly concave. Let I = {H > -∞}, which is an interval, whose endpoints will be denoted by a = inf I ∈ [0, 1] and b = sup I ∈ [0, 1], with a < b to avoid trivialities. We extend H on I c by setting H ( ) = +∞ for ∈ [0, a]\I , and H ( ) = -∞ for ∈ [b, 1]\I . (b) For every ∈ (a, b), H ( ) < 0. (c) H is analytic on (a, b). (d) Either H(a) = -∞ or H(a) = lim y a H(y). The same for b.

  .1.25) (b) (Bounds for Q in (a, b)) For any [u, v] ⊂ (a, b), there exists c 1 , c 2 > 0 such that

  .1.26) (c) (Upper bound for Ψ) For any B, δ > 0, there exists c > 0 such that for every N m 1, Ψ(m, N ) cN c e N H( m N )+δN + cN c e -BN . (4.1.27)

  Example 4.1.5. With Stirling's approximation, we get the following estimates of H and Q for each of the three models of Example 4.1.2.1. (Overtwist and supercoils)H os ( ) = sup ς∈[0, ] ψ(ς, ) = ψ(ς 0 ( ), ),(4.1.29)whereψ(ς, ) = ςwlog 2χ (1 -ς) 2 + logς log(ς) -(ς) log(ς). (4.1.30)And one can derive the asymptotic behaviour of Q:Q os (m, N ) ∼ q m N , where q( ) = ς 0 ( )(ς 0 ( ))|∂ 2 ς ψ(ς 0 ( ), )| . (4.1.31) The function ψ is concave on the convex domain {0 ς

Proposition 4 . 2 . 10 ( 16 )

 421016 any δ > 0, µ n (|X n -c| δ) -----→ n→+∞ 0. Laws of large numbers for the contact density and the largest jump). For any h ∈ R, Remark 4.2.11. When c = 0 (i.e., E[η] = +∞), the limit in (4.2.16) becomes 1 =0 .Before discussing what it tells us about typical paths of the kernel models, let us give a refinement about the largest jump(s) behaviour. Recall the definition of a N (Lemma 4.A.4), which is roughly N

Proposition 4 . 2 . 13 (

 4213 Fluctuations of the contact density). If h < h < h , then |τ | has Gaussian fluctuations: there exists σ > 0 (explicit) such that |τ |

Remark 4 .

 4 2.14. For h ∈ (h , h ), we only gave the law of large number for |τ | and M |τ | .

Proposition 4 . 2 . 22 (

 4222 Critical regime, α > 2: upside deviations). Suppose N LE[η] and let us write

2

 2

Figure 4 . 2 :

 42 Figure 4.2: A schematic view of the localized phase for the kernel models, completely similar to the PS localized phase. The right line is the defect line. On the right, a zoom show the local structure of the renewal: all jumps are typically finite, the largest jump has size O(log N ).

Figure 4 . 3 :

 43 Figure 4.3: Delocalized phase: only a few sites (o(N )) are pinned near the the extremities.

Figure 4 . 4 :

 44 Figure 4.4: Big jump phase: a unique large loop emerges, its size is proportianal to N . The rest of the renewal is "typical" and the other loops have finite length (more precisely: the largest (once the big jump is excluded) has a power law size, which just reflects the power law behaviour of K(n) for n → ∞).

4. 3 . 1 Free

 31 Energy g( ) Proof of Lemmas 4.2.1.

Lemma 4 . 3 . 3 .

 433 For any h ∈ R, the function → h + U ( ) reaches its maximum at a unique maximum point (h) ∈ I \{1}. The function h → (h) is continuous and nondecreasing, with (h) = a for h h , (h) = b for h h , and (h c ) = c . The function (h) is analytic as soon as h ∈ C . The free energy f H is analytic on R\C .

( 4 . 4 . 74 )-→ 1 . 2 .

 447412 Note that x L B L → +∞ andLe -x L B L ϕ(B L ) x L B α+1 L (4.4.75) Thus LP x L (x L (η -B L ) t) -→ +∞ te -s ds = e -t . (4.4.76) For the second largest jump when h < h c , consider the eventB = {||τ | -N | N } ∩ a -1 |τ | M |τ | x . (4.5.22) With the same method as before and with Proposition 4.2.16, we get ξ Ψ,h N (B) → ψ(x) (a nontrivial cumulative distribution function). In other words, a -1 |τ | M |τ | converges in distribution to a non-trivial random variable. Then a -1 N M |τ | = a |τ | a N a -1 |τ | M |τ | . (4.5.23)

Proposition 4 . 5 . 2 .Remark 4 . 5 . 3 .

 452453 Suppose h < h < h and define σ = σ(h) such that 1 σ(h) 2 = -g ( ) -H ( ),(4.5.24) (at c take g ( c ) = lim c g ( )) andπ N = 1 p µ N, ,wherep = P N + √ N σN ∨ 1 = ,(4.5.25)(N is a standard gaussian). Thenξ Ψ,h Nπ N t.v.-→ Here the total variation distance can be written as a supremum over the Borel sets:ξ Ψ,h Nπ N t.v. = sup C∈B(∪ L 1 N L ) ξ Ψ,h N (C)π N (C) . (4.5.27) Proof. Here ∈ (a, b) and it satisfies h + g ( ) + H ( ) = 0. Fix > 0 such that a <and + < b. We already know (Lemma 4.5.1) that Z N ∼ ẐN , where ẐN is Z N with the summation restricted to |m -N | N . Here we first and mainly want to prove that if B is large enough, ẐN ∼ T N := |m-N | B √ N log N e mh Ψ(m, N )P(τ m = N ). (4.5.28) We start by getting an upper bound on ẐN -T N . Thanks to Taylor-Lagrange inequality, there exists A > 0 such that for every 1 m N , mh + N H m N N ( h + H( )) + (m -N )(h + H ( )) -A (m -N ) 2 N = N (f H (h)g( )) -(m -N )g ( ) -A (m -N ) 2 N .

( 4 . 2 N 2 =

 422 5.29) When c , g( ) = g ( ) = 0 and set x = 0. When > c , let x > 0 such that E x [η] = 1 . In both cases, remember that g( ) = x + log E[e -xη ] and g ( ) = log E[e -xη ] (see the proof of Lemma 4.2.1). We getmh + N H m N N f H (h) -A (m -N ) 2 N -N xm log E e -xη . (4.5.30)Recall thate -N x-m log E[e -xη ] P(τ m = N ) = P x (τ m = N ) 1,(4.5.31)to obtain, for every 1 m N ,e mh e N H( m N ) P(τ m = N ) e N f H (h) e -A (m-N ) 2 N . (4.5.32)We know (Assumptions 4.1.4) that for |m -N | N , Ψ(m, N ) cN c e N H( m N ) ,(4.5.33)for a certain c > 0. Thus we get the upper boundẐN -T N |m-N |>B √ N log N cN c e N f H (h) e -A (m-N ) cN c+1-AB 2 e N f H (h) .(4.5.34)Now we estimate ẐN . The function H is analytic in a neighbourhood of . Thus, uniformly in m satisfying |m -N | B √ N log N , one has mh + N H m N = N ( h + H( )) + (m -N )(h + H ( )) + (m -N ) 2 2N H ( ) + O (m -N ) 3 N N ( h + H( )) + (m -N )(h + H ( )) + (m -N ) 2 2N H ( ) + O(N -1/4).

( 4 . 5 . 35 )

 4535 With the same computation as before (see (4.5.32)) and then with Assumptions 4.1.4, we gete mh Ψ(m, N )P(τ m = N ) ∼ Q(m, N )e N f H (h) e (m-N ) 2 2N H ( ) P x (τ m = N ) ∼ Q( N , N )e N f H (h) e (m-N ) 2 2NH ( ) P x (τ m = N ).

( 4 . 5 . 36 )

 4536 We now need to analyze the asymptotic behaviour of P x (τ m = N ).

1 . 2 .H

 12 If h < h c then < c , and x = 0. And we know (see (4.A.46)) that, uniformly in m satisfying |m -N | B √ N log N , P x (τ m = N ) = P(τ m = N ) ∼ N K(N -N E[η]). (4.5.37) If h = h c and Var(η) = +∞, then = c , and once again x = 0. With Proposition 4.A.8, and recalling that b N √ N , we have e (m-N ) 2 2N H ( ) P(τ m = N ) ∼ g α ( ) , (4.5.38) uniformly in m satisfying |m -N | C √ N . And lim sup b N e (m-N ) 2 2N H ( ) P(τ m = N ) e C 2 H ( ) sup |g α |, (4.5.39) (H ( ) < 0), uniformly in m satisfying |m -N | C √ N . Combining these two lines, we get sup |m-N | B √ N log N b N e (m-N ) 2 2N H ( ) P(τ m = N ) -g α (0)

( 4 . 5 . 40 ) 3 .- 42 )

 4540342 If either h = h c and Var(η) < +∞ or h > h c . Recall σ 2 x = Var x (η). One has, still with Proposition 4.A.8,P x (τ m = N ) ∼ e 4.3.1) that g ( ) = -( 3 σ 2 x ) -1 .With the same reasoning as before we obtainsup |m-N | B √ N log N 2πσ 2 x N e (m-N ) 2 2N H ( ) P x (τ m = N )e (m-N ) 2 2N (H ( )+g ( ))Recall that g ( ) = 0 in the first two cases (Lemma 4.3.1). Therefore we get in the three casessup |m-N | B √ N log N R(N ) -1 e (m-N ) 2 2N H ( ) P x (τ m = N )e where R(N ) is regularly varying with N : R(N ) = N K(N (1 -/ c )) in the first case, R(N ) = gα(0) 1 α b Nin the second, and b N = (2π N σ 2x ) -1/2 in the third one. Recall that H ( ) + g ( ) = -σ -2 . We obtainẐN ∼ Q( N , N )e N f H (h) |m-N | B √ N log N e (m-N ) 2 2N H ( ) P x (τ m = N ) ∼ R(N )e N f H (h) Q( N , N ) |m-N | B √ N log N e -(m-N ) 2 2σ 2 N ∼ R(N )e N f H (h) Q( N , N ) √ 2πN σ |m-N | B √ N log N P( N + √ N σN = m). ∼ R(N )e N f H (h) Q( N , N ) √ 2πN σ. (4.5.44) With the upper bound (4.5.34) and the estimation (4.5.44) we get the asymptotic equivalence (4.5.28) (as soon as B is large enough). Moreover, if I = r ∈ N * x √ N r -N y √ N , and if A is an event for the renewal, then ξ Ψ,h N (A, L ∈ I) = 1 Z N r∈I e rh Ψ(N, r)P (A, τ r = N ) = 1 Z N r∈I e rh Ψ(N, r)P (τ r = N ) µ N,r (A).

( 4 . 5 . 45 )

 4545 We got in the previous lines a uniform asymptotic equivalence for e rh Ψ(N, r)P (τ r = N ):e rh Ψ(N, r)P (τ r = N ) ∼ R(N )e N f H (h) Q( N , N ) √ 2πN σ P ρN + √ N σN = r ∼ ẐN P ρN + √ N σN = r ∼ Z N P ρN + √ N σN = r .

  h N (A, L ∈ I) ∼ r∈I P ρN + √ N σN = r µ N,r (A). (4.5.47)The last expression is precisely π N (A, L ∈ I). Remark 4.5.4. From Proposition 4.5.2 and the result of Subsection 4.2.3, dealing with the size of the largest jump for the constrained PS model µ N,L (L fixed) in the different regimes, fluctuations results for the largest jump M |τ | in the kernel models can be derived when h ∈ (h , h ). It would be cumbersome to detail all the cases. But let us give one example: when h < h < h c ,

Lemma 4 .A. 1 ( 1 ) 2 ) 1 0 3 ) 1 α

 4112131 Uniform convergence Theorem [14, Th. 1.2.1]). For any 0 < a < b < +∞, sup Lemma 4.A.2 (Potter's bound, [14, Th. 1.5.6]). If ϕ is slowly varying then for any chosen constants A > 1, δ > 0 there exists X = X(1, δ) such that for any x, y X, Lemma 4.A.3. Let g : (0, +∞) → C be a piecewise continuous function such that the integrals +∞ 1 |g(y)|y δ dy and |g(y)|y -δ dy are finite, for some δ > 0. Then x Lemma 4.A.4 ([14, Th. 1.5.12]). There exists a sequence a L , regularly varying with index

1 . 2 .

 12 3.].Lemma 4.A.5. The following behaviours hold as x 0. The function ϕ is the slowly varying function of the renewal and ϕ 1 is, in each integer case, an other slowly varying function.If α = 0 then E[e -xη ] = 1ϕ 1 (1/x)(1 + o(1)), E[ηe -xη ] = ϕIf α ∈ (0, 1) then E[e -xη ] = 1 -1 α Γ(1α)x α ϕ(1/x)(1 + o(1)), E[ηe -xη ] = Γ(1α)x α-1 ϕ(1/x)(1 + o(1)).

( 4 .A. 6 )-

 46 If α ∈ (0, 2)\{1}, then b L = a L . If α ∈ {1, 2} with Var(η) = +∞ then b L is regularly varying with index 1 α . Furthermore b L √ L and b L a L . If Var(η) < +∞, then b L = L Var(η) a L .

Proposition 4 .

 4 A.8 (Local limit theorem). For any α > 0,sup k∈N b L P(τ L = k)g α k -

E

  [e it(τ L -LE[η]) ] = E[e it(η-E[η]) ] L . (4.A.19)One hasE e it(η-E[η]) = xa L n=1 K(n)e it(n-E[η]) n 1+α (e it(n-E[η]) -1it(n -E[η])).

e

  ity -1ity y 1+αdy .

( 4 .

 4 A.21) Therefore the convergence of Proposition 4.A.7 is proved. Then we need to prove the local version. Letf (t) = E[e itη ], ψ(t) = E[e it(η-E[η]) ],(4.A.22) and v α (t) = E[e itXα ], g α (z) = 1 2π R e -itz v α (t)dt. (4.A.23)

Lemma 4 .A. 9 . v α ∈ L 1 .

 491 Lemma 4.A.10. There exists > 0 and c > 0 such that for any t ∈ [-, ],|f (t)| e -c|t| α ϕ(1/|t|) . (4.A.24)First observe that2πP(τ L = k) = π -π e -ikt E[e itτ L ]dt = π -π e -ikt f (t) L dt = π -πe -i(k-LE[η])t ψ(t) L dt.

( 4 . 28 )

 428 A.25) With the change of variable u = ta L we get2πa L P(τ L = k) = πa L z = z k,L = k-LE[η] a L . Then we can write 2π (a L P(τ L = k)g α (z)) = I 1 -I 2 + I 3 + I 4 ,We start with I 3 . Fix ∈ (0, 1) such as in Lemma 4.A.10. Then, for |u| a L , -c|u| α Lϕ(a L ) a α L ϕ(a L /|u|) ϕ(a L ) .

( 4 .A. 29 ) 4 |

 4294 The Potter's bound (Lemma 4.A.2) ensures that, even if it means taking smaller, one has, for L sufficiently large and A |u| a L , ϕ(a L /|u|) ϕ(a L ) 2|u| -α 2 . (4.A.30) On the other hand, if L is sufficiently large, Lϕ(a L ) 2πa L r L . (4.A.34) Fix δ > 0 and let A be large enough. We have |I 2 | |u| A |v α (u)|du δ, 2π |a L P(τ L = k)g α (z)| 2δ. (4.A.39)

z 1 -

 1 |v α (t)| = exp +∞ 0 e ity -1ity y 1+α dy = exp +∞ 0 cos(ty) -1 y 1+α dy = exp (-c|t| α ) , 1+α dz. Thus, v α is integrable.Proof of Lemma 4.A.10. First we estimate 1f (t):1f (t) = e iy sgn(t) y 1+α dy.

( 4 .

 4 A.41) H2: If E[τ 1 ] < +∞ then ∂ h f h (0, h)

Figure 5 . 1 :

 51 Figure 5.1: The case E[τ1] = +∞ (item H1): The function h → f(h) is continuously differentiable and g is analytic on (0, 1).

Figure 5 . 2 :

 52 Figure 5.2: The case E[τ1] < +∞ (item H2): The function h → ∂ h f(h) is discontinuous at h = 0 and g is not analytic at c = ∂ + h f(0).

Figure 5 . 3 :

 53 Figure 5.3: The case β > 0: the free energy f(β, •) is continuously differentiable.

2 .

 2 f H (β, •) is convex and infinitely differentiable on R, with∂ h f H (β, h) -→ h→-∞ a and ∂ h f H (β, h) -→ h→+∞ b.

Proposition 5 . 2 . 3 .∂

 523 For any h ∈ R, one has, P-almost surely, the convergence in probability of contact density: for the disordered PS model|τ | N ν h,ω,β N -----→ N →+∞ ∂ h f(β, h) ∈ [0, 1), h f H (β, h) ∈ (a, b).

Theorem 5 . 2 . 4 ( 1 .

 5241 No big jump for the fixed contact number and Kernel models). Fix β > 0. For every ∈ (0, 1], 5.2.4 should be compared with Theorem 4.2.15 and with Proposition 4.2.10: the disorder has wiped out the big jump transition! This contrasts with[START_REF]Denaturation of circular DNA: Supercoils and overtwist[END_REF] § IV]

  Notations. Commençons par préciser les notations utilisées pour le modèle d'Ising. Le Hamiltonien associé au modèle d'Ising sur un graphe * non orienté G = (V, E) est donné par H(σ) = -1} V , (1.1.1) où h x est le champ magnétique externe au site x, et J x,y est le coefficient de couplage associé à l'arête (x, y). À une température inverse β, la probabilité d'une configuration σ On parle de modèle d'Ising homogène lorsque le réseau G est régulier (typiquement Z d ), et lorsque le champ externe et les coefficients de couplage sont déterministes et spatialement homogènes (c.-à-d. ne dépendent pas du point x). Il s'agit d'un modèle mathématique du ferromagnétisme. Le modèle d'Ising simplifie substantiellement le fonctionnement d'un métal ferromagnétique puisqu'il suppose que :

	(x,y)∈E
	est	1 Z β	e -βH(σ) .	(1.1.2)

J x,y σ x σ y -x∈V h x σ x , σ ∈ {-1,

  Exemple 1.2.20. La propriété de contraction est en particulier satisfaite lorsque la matrice M est déterministe et admet une unique valeur propre de module maximal, et que celle-ci est une valeur propre simple et réelle. En particulier, donnons des matrices aléatoires V k , bornées, et telles que sup k 1 V k

	aléatoire	1/k L ∞ soit fini. Alors l'exposant de Lyapunov de la matrice +∞ 2 0 + 0 1 k=1

.2.19) muni de la norme • E(M ) . Théorème 1.2.19 ([39]). Soit une matrice aléatoire contractante M , telle que E[|log M |] < +∞. L'application A → L(M +A) est analytique réelle dans un voisinage de 0 dans l'espace de Banach E(M ).

  2.19 de Ruelle garantit que → L Z ( ) est analytique sur R\{0}. Ne reste donc que l'étude de L Z ( ) lorsque 0. Lorsque le support de Z est [0, 1δ] pour un δ ∈ (0, 1), le cone {(y 1 , y 2 ) | |y 2 | y 1 } est contracté par M 0 , et, appliquant une nouvelle fois le Théorème 1.2.19, il vient que dans ce cas L Z est aussi analytique au voisinage de 0. En dehors de ce cas on s'attend à ce que L Z ait un comportement singulier en 0.

  Dans[START_REF] Zanon | Weak disorder expansion of Liapunov exponents in a degenerate case[END_REF], B. Derrida et N. Zanon considèrent la limite en désordre faible d'une perturbation d'une matrice diagonale déterministe ayant une valeur propre dégénérée. Une première étape est d'étudier une perturbation de la matrice identité 2 × 2 de la forme

		Pour l'heure, aucun encadrement plus précis n'a été démontré. Et malheureusement,
	aucun argument, même heuristique, ne permet d'accéder précisément à un équivalent
	C Z	2α .
	1.4 Modèles en désordre faible
		Dans cette section nous nous intéressons à des limites en désordre faible des exposants
	de Lyapunov. Cette digression est motivée par un article récent de Comets, Giacomin
	et Greenblatt [26] où est étudiée une limite continue et exactement soluble du modèle de
	Derrida-Hilhorst (sous-section 1.4.2). Un fait remarquable de cet article est que dans la lim-
	ite continue, l'exposant de Lyapunov exhibe un comportement exactement similaire à ceux
	de la conjecture 1.3.2, singularité incluse. On commence par un résultat général de Derrida
	et Zanon, et à une version mathématique de Sadel et Schulz-Baldes, qui s'intéressent à des
	perturbations de la matrice I 2 (sous-section 1.4.1), dans le cadre duquel peut s'inscrire le
	modèle de Comets, Giacomin et Greenblatt.
	1.4.1 Perturbation faible de la matrice I 2
				et (1.3.18)), au
	lieu d'être englobées dans un O( 2β E[X β ]), peuvent être précisément encadrées et fournir
	en définitive un encadrement
		2 α	R( ) c 2α .	(1.3.47)
	Cet encadrement amène plusieurs remarques. Déjà il est compatible avec la prédiction
	2α et la borne supérieure est même du bon ordre de grandeur. Mais aussi, la prépondérance devant 2 α nous dit que le développement (1.3.42) ne peut être poursuivi à l'ordre suivant C Z
	et que R( ) est nécessairement une singularité en 0 (rappelons que due à la parité de la
	fonction → L Z ( ), seule les puissances paires de sont non-singulières).

  Si pour tout η = 0 assez petit, la matrice ηV 1 + η 2 K est irréductible (c'est-à-dire qu'il n'existe pas de direction déterministe qui soit p.s. stable par la matrice), alors il existe des fonctions ρ k ∈ C ∞ (P 1 ) telles que pour toute mesure invariante ν η et pour toute f ∈ C ∞ (P 1 ),

	Théorème 1.4.1.
	.4.3)
	On suppose en outre que n 2, et que les coefficients de chacune des matrices aléatoires
	V k et W η sont bornés. Enfin on pose K := E[V 2 ]. Ce cadre réduit permet d'obtenir (1.4.2)
	(et davantage) sous des hypothèses moins restrictives que celles requises par le résultat
	de [86]. Le théorème suivant donne une condition pour que les mesures invariantes de M η
	sur P 1 admettent (au sens faible) une décomposition en puissance de η.

  .4.7) (W désignant une variable aléatoire centrée), ils obtiennent une expression de λ 2 en terme d'une dérivée logarithmique, respectivement, d'une fonction de Bessel de première espèce I ν , de seconde espèce K ν , ou de la fonction d'Airy Ai.

	1.4.2 Limite continue du modèle de Derrida-Hilhorst : un modèle ex-
	actement soluble
	L'idée de F. Comets et al. [26] est de considérer une limite continue du produit de
	matrices aléatoires M n, • • • M 1, . Plus précisément une limite continue du processus de Markov

  A.3) où est la mesure de Haar sur l'espace projectif P 1 , et les fonctions ρ k sont des fonctions C ∞ qu'il convient de déterminer. Pour définir une mesure invariante, la fonction densité ρ η doit satisfaire M * η ρ η = ρ η (où M η est l'adjoint, au sens L 2 ( ), de M ). Écrivons un développement formel de cette identité par rapport à η afin de déterminer les fonctions ρ k .Observons d'abord que l'opérateur M η admet un développement par rapport à η :

  Démonstration du Théorème 1.4.3. D'après le Théorème 1.2.5, il existe une mesure invariante ν η telle que

.A.22) Les équations (1.A.15) et (1.A.22) fournissent une preuve de l'hérédité. Le théorème est démontré.

  The Assumptions 2.1.5, together with a convexity argument, ensure that A is an interval of positive length. Note that α = +∞ if and only if Z 1 almost surely. In any case A takes the following form: either E[Z

		.1.9)
	and
	α = sup A ∈ (0, +∞].	(2.1.10)

α ] = 1 and then A = (0, α), or E[Z α ] < 1, and then A = (0, α]. In the latter case, necessarily, E[Z γ ] = +∞ for every γ > α. Here is the main result of this work.

Theorem 2.1.6. There exist positive coefficients ( k ), where k is a rational function of the moments E[Z], . . . , E[Z k ], such that the following expansions hold, as 0.

  Remark 2.5.4. This remark provides a short explanation for the claims of Remark 2.1.7, items 4 and 5. In both case, only the upper bound on R( ) requires a specific proof. Recall what was mentioned in Remark 2.3.2: the regular part is derived up to and including order 2( α -1) and the upper boundβ E[X β ] can be replaced by E[( 2 X ) α ∧ ( 2 X ) α ].We only need to prove that the upper bound is O( 2α ) for Remark 2.1.7 item 4, and o( 2α ) for the item 5.

	).	(2.5.10)

  .A.16) Remark 2.A.6. One could be surprised that the upper bound involvesE[1 + X β ] instead of E[ X β ]. Such a caution was not necessary in the previous context since the latter was bounded form below as goes to 0. Here, a priori, it could happen that E[ X β ] vanishes as goes to 0.

	Remark 2.A.5. For Proposition 2.A.4 to be usable, one needs to control E[ X β ]. With
	Lemmas 2.A.2 and 2.A.3, one has E[ X β ] = O(1) as goes to 0 as soon as (2.A.4) holds.
	Remark 2.A.7. The existence of G (l) , for l	K, is ensured by the assumption (2.A.15),
	which gives lim →0

  .2.8)As said, the point h c corresponds to the big jump/localization phase transition whereas h and h (when they are finite) correspond to other types of phase transition, which will be briefly discussed in Subsection 4.2.4. First let's see what happens for the three circular DNA models.Example 4.2.5. In the models of Example 4.1.2, H has infinite derivative on the boundary of its domain. Thus h = -∞ and h = +∞. Only the critical point h c can exist. We detail it for each of these models.

	1. (Overtwist and supercoils

  If u L → +∞, then (4.2.20), (4.2.21) and (4.2.22) hold. Remark 4.2.21. We leave open the question for u L → -∞. Note that then x L → 0 (see Def. (4.1.22)) and there exists an explicit sequence A L → +∞ such that P x L (x L (M L -A L ) t) -----→

	w ] = exp	0	w	e ity -1 -ity y 1+α	dy .	(4.2.28)
	2. L→+∞	e -e -t .	(4.2.29)

  .2.30) which defines u L 1. If u L → +∞, then (4.2.20), (4.2.21) and (4.2.22) hold.

  Proof of Proposition 4.2.18. First note that x L → x as L → ∞, where x solves E x

	.4.37)
	4.4.2.3 Supercritical regime (Proof of Proposition 4.2.18)

  .4.66) 4.4.2.5 Critical behaviour for η ∈ L 2 (Proof of Proposition 4.2.23) Proof of Proposition 4.2.23. Note that, with Lemma 4.A.5, as x 0.

		E x [η] = E[η] -xσ 2 (1 + o(1)).	(4.4.67)
	Thus, x L satisfies	x L ∼ y L =	θ L σ 2 L	.	(4.4.68)

Remerciements

Main Results

We discuss the main results about the constrained PS model µ N,L (its free energy and how the conditioning τ L = N affects the size of the largest jump), and detail the consequences on the kernel model (free energy and path properties). 1. The function → g( ) is continuous on [0, 1] and continuously differentiable on [0, 1); 2. g( ) = 0 for ∈ [0, c ] and it is negative, decreasing, concave, and analytic on ( c , 1],

Free Energy

with g ( ) → -∞ when → 1.

Remark 4.2.2. Recall that c = 0 when E[η] = +∞, in particular when α < 1.

The next lemma gives the critical exponent of the free energy g for what will be the big jump/localization phase transition. where φ is a slowly varying function (details in Lemma 4.3.2), and 1)) (4. 3.14) with φ(u) → 0 when α = 1 and φ(u) → +∞ when α = 2 as u → +∞. Thus,

Therefore, with [14, Th. 1.5.12], x ∼ δ 1 α-1 ϕ 0 (1/δ) for a certain slowly varying function ϕ 0 as long as α = 1.

(4. 3.16) When α = 2, ϕ 0 (u) → 0 as u → +∞.

When α = 1, we have φ(1/x ) ∼ E[η] 2 δ (4. 3.17) with φ(u) → 0. We know [START_REF] Bingham | Regular variation[END_REF]Prop. 1.3.6], that for any a > 0, x a ϕ(1/x) as x 0. So, for any a > 0, x δ with φ(u) → 0 when α = 0 and φ(u) → +∞ when α = 1 as u → +∞. And 1/ = δ -1 . Therefore, with [14, Th. 1.5.12], x ∼ δ 1 1-α ϕ 0 (1/δ) for a certain slowly varying function ϕ 0 as long as α = 1. And g ( ) ∼ -x α φ(1/x ) ∼ -δ α 1-α φ(1/δ).

(4. 3.21) For α = 1 the discussion is the same as before since 1 φ(1/x ) ∼ δ, with 1 φ(u) → 0.

When α = 1, to obtain the asymptotic behaviour of g from what we found for g , we use [14, Prop. 1.5.10].

Free Energy f H (h)

The proof of Lemma 4.2.4 is postponed to section 4.5 (see the proof of Lemma 4.5.1). Until there, f H will be the function

{ h + g( ) + H( )}. ). The method is the same, only the form of the expansions of U and U change. Set again =c . We work in the limit 0. Recall the critical behaviour of g and its derivative when > 0:

g( ) = -κ φ(1/ ), g ( ) = -κ κ-1 φ(1/ )(1 + o(1)). (4.3.41) Write the expansion of H and H :

where k + 1 = κ and a 2 > 0. Recall also that H ( c ) = -h c . Thus, when κ ∈ N,

).

(4.3.45) In both cases it can be written

with ψ(1/ ) = o(1/ δ ) for any δ > 0. Note that when κ ∈ N, ψ = a κ + φ is not necessary slowly varying for it can vanish. Similarly,

with a κ = 0 when κ ∈ N. Then the optimizer ˆ = ˆ (h), which solves h + U ( ) = 0, has an expansion:

where g = h-h c . ψ does not necessarily slowly vary, but it satisfies ψ(1/g) = o(1/g δ ) for any δ > 0. By injecting it into the equation h+U ( (h)) = 0 we get the coefficient

.50) where T is a polynomial. To conclude, just note that the same method for We start by what is the main result of [START_REF] Armendáriz | Thermodynamic limit for the invariant measures in supercritical zero range processes[END_REF], whose proof can be adapted to cover some cases in the critical regime. Recall Lemma 4.A.11: suppose E[η] < +∞ and let c N be such that

Denote by π N,L the distribution, under P, of

where s is the random permutation that exchanges the L th coordinate with a uniformly chosen coordinate among {1, . . . , L}.

Proposition 4.4.1. Let c N be such that † c N b N and such that (4.4.2) holds, with

where • t.v. is the total variation distance: Proof. Let T be the map that exchanges the biggest coordinate and the last one: b L (and in particular M L = ηL under πN,L ). Eventually, observe that

Proof in the other regimes

General strategy and notations

We use the same strategy as [4], starting from the elementary formula

When the density L N is sufficiently large, we will rather use the tilted measure and the identity

with x L such that E x L [η] = N L . Then, using that

the first step is to find c L → +∞ such that LP x L (η > c L ) is bounded, and bounded away from 0. The second (and a bit harder) step is to prove a local limit theorem to estimate

Before going through the proofs, let's introduce some notations. Besides the tilted measure P x , we will use

The superscript c L will always be omitted. Under these laws, the renewal distribution is respectively

The next lemma (which already appears in [4]) allows asymptotic estimates of

and Var(η) < +∞. Observe that x L → x which solves E x [η] = 1 , and that σ 2

x = lim L→∞ Var x L (η). Then,

4.5 Proofs: path properties for the models with a kernel

In this section we set = (h) ∈ [a, b], the maximum point of → h + g( ) + H( ). Recall that

e mh Ψ(m, N )P(τ m = N ). Proof. Fix > 0. First we get a lower bound on Z N . Observe that for any y ∈ (a, b),

On the other hand we try to obtain an upper bound on Z N -ẐN . For any N m 1, Since y → yh + g(y) + H(y) is strictly concave, one can choose δ and small enough, and B large enough, so that K < f H (h), and therefore lim sup

Using the same method, we can obtain, for any δ, B > 0, lim sup

and therefore lim sup

Combining (4.5.5), (4.5.11) and (4.5.13), we get

and lim

The continuity and properties of h → (h) can be found in Lemma 4.3.3. Now look at the largest jump(s).

1. When h < h c , we know that (h) < c . From Lemma 4.5.1, we know that, for any > 0,

e mh Ψ(m, N )P(τ m = N ). (4.5.17)

Consider the event

We compute

and the convergence can be made uniform on {|m -N | N }. Thus,

We get the convergence (4.2.16).

If

with ϕ 1 ϕ and lim +∞ ϕ 1 = +∞.

with ϕ 1 ϕ and lim +∞ ϕ 1 = 0.

). (4.A.9)

with ϕ 1 ϕ and lim +∞ ϕ 1 = +∞.

7. If α 2 and Var(η) < +∞ then

(4.A.11)

4.A.3 Limit Theorems

Recall that

Proposition 4.A.6 (Limit theorem for M L ). For any α > 0, and any t 0,

for sequences µ L , b L and an α-stable random variable X α , which are detailed for each case.

µ L = 0 when α ∈ (0, 1);

123 Hence,

If is small enough, then for every |t| ,

In particular for any < c , sup

Proof. The main references are [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF] and [START_REF] Berger | Notes on random walks in the Cauchy domain of attraction[END_REF], for which we refer to the corresponding results. In this chapter, we consider a disordered version of the fixed contact number and kernel PS models introduced in Chapter 4. Like in the disordered PS model (Section 3.2), the inhomogeneity of the interactions between the two strands (A-T bonds are weaker than G-C bonds) is modeled by IID random variables (ω j ) j=1,2,... , whose law is denoted by P; the disorder is independent of the renewal randomness. For the sake of conciseness, we assume that the disorder is bounded (Hypothesis 5.1.1).

Our main result is that disorder suppresses the big jump phenomenon and we will argue that both at the free energy level and by analyzing the trajectories of the model.

Models and Assumptions

The setting is the same as in Chapter 4, except that the contact interaction h at site j is replaced by βω j + h, where β is a nonnegative parameter (of course the homogeneous model is recovered by taking β = 0). We still work under Assumptions 4.1.3 and 4.1.4, and we make the additional assumption on the disorder: We are interested in the two following models.

Disordered fixed contact number model. For any 1 L N , we consider the random measure

where A ∈ F (see (4.1.5)), δ j = 1 j∈τ , and the partition function is the random variable

Disordered kernel model. The kernel model becomes in presence of disorder:

Once again the partition function can be decomposed depending on the number of contact points:

For simplicity, the results will only be stated under the following assumption, which is fulfilled by the circular DNA mechanisms introduced in Chapter 4 (Example 4.1.2). By taking formally Ψ = 1 (which does not fulfill Assumptions 4.1.4), we recover the standard disordered PS model, defined by

(5.1.7)

and whose partitions function is:

(5.1.8)

Main Results

We investigate two aspects of these models. First the free energy properties (smoothness and critical exponents), second the typical trajectories, depending on the parameters. For the disordered PS model ν h,ω,β N , these have been deeply studied and a summary of the known results can be found in Section 3.2. These results are important for us both for comparison with some of the new results we obtain in this chapter and because we are going to exploit some of them in the arguments of proof.

Free Energy: existence and regularity

The first result gives the existence of the free energies (that is the convergence of the partition functions), and provides variational formulas relating them. Proposition 5.2.1. For any β 0, h ∈ R and ∈ [0, 1], in the case of the fixed contact number model we have, almost surely and in L 1 , the existence of the limit lim N →+∞:

while for the kernel model in the same sense we have the existence of the limit

and we recall that for the standard disordered PS model

The free energies g(β, •), f(β, •) and f H (β, •) are finite, non random, and obey the conjugate variational formulas

and

(5.2.5)

From now on, unless otherwise specified, we suppose β > 0. Our approach is to exploit what is known about f(β, h) to extract information about g(β, ), thanks to the variational formula (5.2.4), and then employ this information to establish results about f H (β, h).

The disorder smooths the fixed contact number model transition. We are now going to explain how the big jump transition mechanism at the free energy level for β = 0 does not exist anymore for β > 0.

We first present the argument in an informal way, carrying along the parallel between β = 0 and β > 0, and then we will give two precise statements. So the point is to argue that from the variational formula

and by using the smoothing bound (3.2.10) one readily obtains that the optimization procedure changes radically when switching from β = 0 to β > 0.

First of all for the homogeneous model (β = 0) the situation is:

is solvable for every ∈ (0, 1): the solution is unique, positive and depends analytically on (Figure 5.1).

Proofs: Free energies

Free energy g(β, )

We start with the existence of g(β, ). It is based on a standard super-additivity argument. Some care is needed though. We simplify the notation Z ω,β N,L = Z N,L,ω and set, for ∈ (0, 1], ZN, N,ω := min

where the set over which the minimum is taken reduces to one point if N is an integer. The basic observation is that

which is obtained by inserting the event that τ n = N and using the independence of the increments of τ . Since (N + M ) is either N + M or N + M we readily obtain that log Z N +M, (N +M ) ,ω log ZN, N,ω + log ZM, M,θ N ω , (5.3.5) and a strictly analogous argument holds for log Z N +M, (N +M ) ,ω , obtaining exactly the same lower bound. Therefore (5.3.2) is proven.

Proposition 5.3.2. For any ∈ [0, 1], there exists a deterministic quantity g(β, ) such that

almost surely and in L 1 . Furthermore, for any ∈ [0, 1), there exists c = c( ) > 0 such that, for any > 0, lim sup

almost surely and in L 1 . For = 1, lim sup

where g(β, 1) = g(1) = log K(1).

Proof. Fix ∈ (0, 1]. First of all, in order to apply Kingman's sub-additive ergodic Theorem we have to manage the fact that ZN, N,ω becomes zero when N < 1. But Kingman's Theorem can be applied to {log ZnN 0 , LN 0 ,ω } n=1,2,... , for N 0 = 1/ , so we can set g(β, ) := lim

where m = (Nδ N ) and δ N is a well chosen sequence. Choose δ N such that Nδ N is a multiple of N 0 and such that for every L satisfying |L -N | N , one has

One readily check that these inequalities are satisfied as soon as

Thus, in addition, we can assume (and we do) that lim sup

.3.14)

Going back to (5.3.11), the first term satisfies, with (5.3.10),

Thus, with the assumption on δ N we get lim sup

The second term can be bounded from below (recall the assumption

ω N -j .

(5.3.17)

With the assumption on δ N , we check that lim sup

Therefore, (5.3.17) yields lim inf

At the end of the day,

20)

P-a.s. and in L 1 . In the same way we get an upper bound:

where m -= (N + δ N ) and m + = (N + δ N ) . In the same way as in (5.3.17), the second term can be bounded from below by some c N , thus yielding lim sup

Combining (5.3.20) and (5.3.22), we get (5.3.7). For = 0, it is easier: when 1 L N ,

and we get (5.3.7), with g(β, 0) = 0. Finally, for = 1, and 0 N -L N ,

(5.3.24)

Then, with the law of large numbers and Lemma 4.2.1, we get P-a.s.,

Similarly,

Thus we obtain g(β, 1) = g(1), as well as (5.3.8).

Lemma 5.3.3. The map → g(β, ) is continuous and concave on [0, 1], with g(β, 0) = 0 and g(β, 1) = g(0, 1) = log K(1), and ∂ g(β, ) → -∞ as → 1.

Proof. We proceed by steps.

-Fix 0 1 < 2 1 and λ ∈ (0, 1). For any N sufficiently large,

(5.3.27) Thanks to the convergence (5.2.1), we get in the limit

(5. As a consequence, for any ∈ (0, 1],

(5.3.30)

Then, With the concavity we get lim 1 ∂ g(β, ) = -∞.

-The continuity of g(β, •) on (0, 1) is a consequence of concavity. The continuity at 1 is a consequence of (5.3.30) and of continuity of g(•) (Lemma 4.2.1). A similar argument yields |g(β, )g( )| Cβ , and thus the continuity at 0.

Free energy f(β, h) and f H (β, h)

Proof of Proposition 5.2.1. The convergence (5.2.1) has been proved in the previous subsection. Let us prove the convergence towards f H (β, h). For f(β, h) it is identical: change Ψ in 1 and H in 0. For any y ∈ (a, b) one has

{hy + H(y) + g(β, y)} = f H (β, h).

(5. 3.35) On the other hand, fix δ, B > 0

(5.3.36) One readily sees, dividing [0, 1] into small segments and using (5.3.7), that P-a.s.,

(5.3.37) The same goes for the second maximum in (5.3.36), and we obtain lim sup

Letting δ → 0 and B → +∞ we get the result:

(5.3.39)

The convergence (5.2.2) is proved. As a supremum of linear functions, h → f H (β, h) is convex. And, since g(β, •) is concave (Lemma 5.3.3), we get the reciprocal variation formula.

Free energy regularity

Recall that β > 0 is fixed. The aim of this section is to exploit the variational formulas (5.2.4) to derive results on g(β, •) from f(β, •)'s properties. The next lemma explains at which point the infimum/supremum in these variational formulas are reached. Lemma 5.3.4.

For any

2. The function g(β, •) is strictly concave and continuously differentiable. For any ∈ [0, 1), the point h(β, ) := -∂ g(β, ) is the only point where the infimum inf h∈R {f(β, h) -h} is reached.

The function f H (β, •) is continuously differentiable and, for any

Proof of Lemma 5.3.4 and Theorem 5.2.2. Fix β > 0.

-We start with a basic observation: since f(β, h) = 0 for h h c (β), on has, for

(5.3.40)

The function

We know (Theorem 5.5.1) that ∂ 2 h f(β, h) > 0 for every h > h c (β), and that f(β, •) is C ∞ on (h c (β), +∞) (Theorem 3.2.2). Therefore, by the implicit function theorem, the function

which is thus infinitely differentiable on (0, 1), with ∂ g(β, ) = -h(β, ). And g(β, •) is strictly concave on (0, 1).

-As for the supremum

it is reached when f(β, h)h = g(β, ), that is, with the previous item, when

-If f(β, •) is analytic on (h c (β), +∞), then so is ∈ (0, 1) → h(β, ) ∈ (h c (β), +∞). And therefore g(β, •) (and similarly f H (β, •)) is also analytic.

-With the same method, we obtain item 3 and f H (β, •)'s properties.

Eventually, observe that since ∂ h f(β, h) is positive when h > h c (β) and vanishes as h h c (β), we have

as soon as c δ sufficiently small. Proposition 5.3.5 (Behaviour of g(β, •) at small density). There exists C > 0 such that for any sufficiently small,

) when 0, for some slowly varying function φ and γ 2, then g(β,

where ϕ 1 also slowly varies.

Proof of Proposition 5.3.5. With (5.3.43), for sufficiently small,

Then, when is sufficiently small, inf

These are easy to compute and one gets

At the end of the day,

If f(β, h c (β) + ) c 2 , for small (Theorem 3.2.2), then

(5.3.52) for small.

Remark 5.3.6. When Assumptions 5.1.2 are not fulfilled, the same phenomenon as for the homogeneous model emerge: the continuous differentiability is only true (a priori) on an interval (h (β), h (β)), where

The behaviour of f H (β, h) when h approaches h (β) or h (β) can be deduced from the behaviour of g(β, ) and H( ) as approaches a or b.

Proofs: path properties

Proof of Proposition 5.2.3. We prove it for the kernel model. The proof is the same for the disordered PS model (replace replace H with 0). Fix > 0, h ∈ R and set = ∂ h f H (β, h).

We prove that

where ẐN is Z N with the sum restricted to |m -N | N . Of course Z N ẐN . More precisely, we prove that there exist c > 0, and a sequence η N (ω), which vanishes P-a.s. when N → +∞, such that for every N 1,

Using the same method as in the proof of Proposition 5.2.1, we obtain, almost surely, Then the result is used to obtain a similar result, first for µ ω,β N,L and then for ξ Ψ,h,ω,β

N

. This is achieved by using that ν h,ω,β N and ξ Ψ,h,ω,β N can be written as a mixing of measures µ ω,β N,L .

Proof of (5.2.14). Fix β > 0 and h > h c (β), so that f(β, h) > 0. To simplify, we will omit the superscript h, ω, β for ν h,ω,β N and Z h,ω,β N . Fix δ > 0. First we use an union bound, decomposing the event depending on the extremities of the largest jump

where, for 1 x y N , We know that P-almost surely,

Therefore, P-almost surely,

and finally lim sup

(5.4.17)

In particular we get (5.2.14).

Proof of Theorem 5.2.4, part 1. If h ∈ R, we know that for any event A for the renewal,

Consequently, for any ∈ (0, 1) and > 0, one has

.4.19)

Fix ∈ (0, 1), and h such that ∂ h f(β, h) = . Then, P-a.s.,

and, with (5.3.7), lim sup

(5.4.21)

Thus, P-a.s.,

In particular, for the event A = {M |τ | δN }, we obtain, thanks to (5.4.17), lim sup

. By the proof of Proposition 5.2.3, there exists c > 0 and η N (ω), which vanishes a.s. as N → +∞, such that

On the other hand, for any event A for the renewal,

(5.4.25) Eventually, fix h such that H = ∂ h f(β, h). Therefore, with (5.4.22) and (5.4.24), P-a.s.,

Once again, the result follows from (5.4.17).

On strict convexity for the PS free energy

The proof of the following result stems from a suggestion by Hubert Lacoin.

Theorem 5.5.1. For every β > 0 and h > h c (β),

(5.5.1)

Proof. The argument is close in its basic idea to what one finds in [50, Ch. 5, Sec. 2]), that is the conditioning on what happens on even sites. To simplify the exposition we assume like in [START_REF] Giacomin | Random polymer models[END_REF] that both K(1) > 0 and K(2) > 0: the argument easily generalizes, but at the expense of heavier notations (they are developed for example in [START_REF] Hollander | Random polymers[END_REF]Ch. 7]). We know [56, Proof of Th. 2.1] that

So it suffices to find a lower bound on the variance: without loss of generality, let us make this estimate with N replaced by 2N . To simplify the notations we set ν 2N,ω := ν h,ω,β 2N : the disordered PS measure. We start by introducing the σ-algebra F e generated by δ j for j even. Let us remark that, with Jensen's inequality,

(5.5.3) Now we consider the conditional variance on the event E σ := {τ | δ 2j = σ j } for every given configuration σ ∈ {0, 1} {0,1,...,N } , and we can of course restrict to σ 0 = σ N = 1. Given a configuration σ, we set n(σ) (the number of even pinned sites), and we introduce, for j = 0, 1, . . . , n(σ) the even contact locations by setting 0 = 0 and j+1 = 2 min{ > (5.5.9)

Therefore it suffices to find a lower bound on lim inf

N →+∞

2N

N -1 k=0 EE ν 2N,ω δ 2k δ 2(k+1) .

(5.5.10)

We claim that, for k = 0, . . . , N , 5.5.11) and in the same spirit, a lower bound which will be used later:

where we used the regular variation of K to obtain that sup j 3 K(j)/K(j -2) is finite. Thus, The term j = 1 also deserves a special treatment and we need to sum on every possible second jump: Bibliography