Jean-Baptiste Bardet

Jean-Yves Brua

Gaëlle Chagny

Antoine Channarond

Olivier Guibé

Francky Luddens

Keywords: Métastabilité, Blume-Capel, Configurations Critiques, Temps de Transition, Théorie du potentiel vii Metastability, Blume-Capel, critical configurations, transition times, potential theory viii

La rédaction des remerciements est un exercice dont je n'avais pas anticipé la complexité. C'est le regard fixé sur sa feuille blanche ou son écran que l'on passe la grande majorité des années de thèse, et c'est seulement à la fin que l'on prend le temps de faire le bilan du chemin parcouru. Je me souviens de tous les bons moments, j'apprécie à leur juste valeur les leçons que les moins bons m'ont enseignées, mais surtout je réalise à quel point mes chances de réussite auraient été proches de zéro sans le soutien de nombreuses personnes pour m'épauler. Il est inconcevable pour moi de ne pas les remercier.

Je remercie tout d'abord Claudio Landim. Ces années passées sous sa direction m'ont permis de voir de près pourquoi la simple évocation de son nom suffit aux spécialistes, et même à d'autres, à garantir la pertinence et la précision des résultats cités. Son expérience, son savoir, et son expertise sont inestimables pour quiconque est intéressé par la recherche en mathématiques, et la thèse présentée ici lui doit énormément.

J'ai tant à dire sur Mustapha Mourragui que j'appréhende d'en oublier. Mustapha me suit depuis six années maintenant ; après avoir rédigé mon mémoire de Master 1 sous sa direction, le bilan était si bon pour moi que je n'ai pas hésité une seconde avant de le démarcher pour celui du Master 2, et voici où nous en sommes. Je tiens à le remercier avant tout pour sa disponibilité et son implication dans sa direction. Il a su être attentif à mon évolution et à mes doutes. Ainsi, notre relation a parfois, souvent, dépassé le rapport directeur-étudiant. Aujourd'hui, c'est en tant qu'ami que j'aimerais adresser mes plus sincères remerciements à Mustapha.

Mes pensées se tournent également vers Olivier Benois. Dès ses enseignements de Licence 3, j'ai pu apprécier sa maitrise, sa grande pédagogie et son souci du bien-être des étudiants. La première année de cette thèse m'aura conforté dans l'idée qu'Olivier est à l'écoute, toujours très respectueux des humeurs des uns et des autres. Les mathématiques ont tant besoin de personnes aussi humaines que lui.

Je voudrais également exprimer toute ma gratitude à Roberto Fernandez et Elisabetta Scoppola pour avoir accepté de lire mon manuscrit et assumer la responsabilité d'en écrire les rapports, en particulier en cette période de l'année où les agendas ne sont pas les plus légers. I would like to express all my gratitude to Roberto Fernandez and Elisabetta Scoppola for accepting to review my thesis and write the reports, especially at this time of the year.

Je suis aussi honoré de compter parmi les membres du jury de ma thèse Oriane Blondel, Alexandre Gaudilliere et Ellen Saada. J'adresse également un remerciement particulier à Pierre Calka qui a accepté de le présider, mais aussi pour ses qualités exceptionnelles qui font de lui un directeur de laboratoire exemplaire

Introduction Introduction

La métastabilité est un phénomène observé dans la nature dans de nombreux modèles physiques, chimiques ou biologiques, en particulier dans des systèmes de particules dans le domaine de la mécanique statistique. Elle peut se définir comme la propriété, pour un système de particules, d'être stable d'un point de vue cinétique, mais instable d'un point de vue thermodynamique. En d'autres termes, elle met en étude l'existence d'états stables pour un système, dans lesquels ce dernier demeure pour une longue durée avant de rejoindre son état d'équilibre sous l'action d'une quantité suffisante d'énergie, appelée énergie d'activation. A titre d'illustration simple, voire simpliste, observons l'évolution d'une bille sur le schéma suivant :

Il est évident qu'en position [START_REF] Barrera | Abrupt convergence and escape behavior for birth and death chains[END_REF], la bille se trouve dans un état stable ; sans aucune perturbation extérieure, elle restera immobile. Cependant, si l'on fournit suffisamment d'énergie pour qu'elle quitte l'état (1) pour rejoindre l'état (2) -appelé critique -, elle rejoindra l'état d'équilibre [START_REF] Beltrán | Metastability of reversible finite state Markov processes[END_REF].

Une autre caractéristique de la métastabilité concerne le temps extrêmement long de transition entre un état stable et l'état d'équilibre. Nous pouvons en observer un exemple frappant avec le diamant. Ce dernier est composé d'atomes de carbone qui, sous cette forme, se trouvent dans un état stable. Une fois l'énergie d'activation fournie à ce système, dans le cas présent provenant naturellement des conditions normales de température et de pression, les atomes de carbone quittent l'état diamant pour rejoindre l'état d'équilibre graphite, mais ce après plusieurs milliards d'années.

La métastabilité est étudiée par bon nombre de scientifiques, qu'il s'agisse de physiciens ou encore de mathématiciens. Bien que les premières études mathématiques de la métastabilité datent du début du XX eme siècle, avec les travaux de Eyring et Kramer, il a fallu attendre les approches de Freidlin et Wentzell [START_REF] Freidlin | Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF] pour observer les premiers résultats obtenus dans les mathématiques modernes. La théorie des grandes déviations permet d'étudier le comportement asymptotique des systèmes 1 métastables. L'approche utilisée est dite trajectorielle, ou pathwise approach dans la littérature.

Au début du XXI eme siècle, Bovier, Eckhoff, Gayrard et Klein (BEGK) étudient la métastabilité sous un angle différent avec le développement de la théorie du potentiel. Le raffinement des résultats déjà obtenus est conséquent, puisque cette méthode permet pour la première fois d'estimer le temps d'atteinte d'états pertinents, en particulier les états critiques, qui constituent l'un des enjeux majeurs dans l'étude de la métastabilité, ainsi que la probabilité d'atteindre le voisinage de certains états métastables. A ce stade, il apparaît comme naturel que la théorie du potentiel présente de nombreux avantages dans l'étude de la métastabilité. En particulier, la notion de capacité se révèle être centrale, et il faut très souvent estimer cette quantité à l'aide des principes de Dirichlet et de Thomson (cf. (2.9) et (2.10)).

Au début des années 2010, Beltran et Landim [START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains[END_REF][START_REF] Beltrán | Metastability of reversible finite state Markov processes[END_REF][START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains II, the nonreversible case[END_REF][START_REF] Beltrán | A martingale approach to metastability[END_REF] proposent une approche de type martingale, développant les travaux de BEGK et parviennent à décrire les comportements métastables de processus sous des hypothèses faibles : on suppose ici que le processus est irréductible, que son état stationnaire est réversible, et que les produits finis de taux de saut sont comparables (cf. [START_REF] Beltrán | Metastability of reversible finite state Markov processes[END_REF] pour plus de détails). Pourtant, leur principal résultat assure de l'existence de différentes échelles de temps sur lesquelles nous pouvons observer un comportement métastable. Bien que ce soit encore une fois l'estimation de capacités, aidée par les deux principes précédemment cités, qui se trouve au centre des démonstrations, la méthode s'appuie également sur une réduction drastique du champ d'étude grâce à l'utilisation du processus-trace, permettant de restreindre l'observation du comportement du processus aux états pertinents, et donc de s'affranchir de données provenant d'évènements négligeables.

Contrairement à l'approche pathwise, celle-ci ne met pas en évidence le chemin précis parcouru par le processus d'un état stable x à un autre état stable y. En revanche, la caractérisation des états critiques permet un calcul exact de la profondeur des puits des états stables, le puits d'un état x étant un voisinage qui, une fois atteint par le processus, le conduit en x très rapidement. Ceci permet de donner une description exacte de la dynamique asymptotique entre les puits, en particulier de montrer l'existence de suites (θ N ) telles que T N /θ N converge vers une variable exponentielle de moyenne 1, où T N est l'instant où le processus quitte un état métastable.

Cette thèse utilise les techniques développées par Beltrán et Landim pour le cas particulier du modèle de Blume-Capel.

Le modèle de Blume-Capel a été introduit en 1966 pour modéliser la transition He 3 -He 4 . Il s'agit d'un modèle de spins à trois états +1, -1, 0, représentant respectivement une particule chargée positivement, négativement, et l'absence de particule, sur un réseau. Bien que considérer Z d comme réseau pour ce modèle ait du sens, cette thèse traite du cas du tore avec conditions périodiques au bord : Λ L = (Z/LZ) 2 . L'espace d'états est donné par Ω L = {-1, 0, +1} Λ L , dont les éléments sont appelés des configurations. Chaque configuration σ ∈ Ω L possède ainsi un certain niveau d'énergie représenté par l'hamiltonien H défini par

H(σ) = x,y∈Λ L ||x-y||=1 (σ(x) -σ(y)) 2 -λ x∈Λ L (σ(x)) 2 -h x∈Λ L σ(x),
où la première somme ne compte les paires qu'une seule fois, λ ∈ R est le potentiel chimique, et h ∈ R est le champ magnétique extérieur. Nous observons alors qu'un tel système est paramétré par λ et h. Plus particulièrement, chaque somme traite d'une quantité pertinente pour le modèle. A l'évidence, la première indique le nombre total d'interfaces d'une configuration σ. Notons que les interfaces de spin +1 et de spin -1 ont une contribution élevée. Plus particulièrement, la seconde somme indique la magnétisation de la configuration σ, c'est-à-dire la valeur moyenne d'un spin de σ, tandis que la troisième somme indique sa concentration, autrement dit le nombre total de spins chargés. Nous considérons dans cette thèse le cas où λ = 0 et 0 < h < 2 afin que les configurations présentant des rectangles de spins 0 soient des minima d'énergie locaux, ce qui s'avèrera être central. Pour β qui désignera l'inverse de la température (que l'on fera tendre vers l'infini), nous considérons une dynamique de Glauber, c'est-à-dire que les spins ne pourront être flippés qu'un par un, dont les taux de sauts sont donnés par : R β (σ, σ ) = e -β[H(σ )-H(σ)] + , σ, σ ∈ Ω L , où [•] + désigne la partie positive. La mesure stationnaire est la mesure de Gibbs associée à l'hamiltonien H :

µ β (σ) = 1 Z β e -βH(σ) ,
où Z β est une constante de normalisation permettant à µ β d'être une mesure de probabilité. En particulier, celle-ci satisfait la condition d'équilibre détaillée

µ β (σ)R β (σ, σ ) = R β (σ , σ)µ β (σ ).
Nous remarquons donc que notre modèle favorise les configurations dont l'énergie est faible ; en particulier, plus le nombre total d'interfaces d'une configuration σ est petit, plus le temps de séjour du processus en cette configuration sera élevé. On observe alors immédiatement que les trois états qui vont dessiner le paysage métastable du modèle sont les états -1, 0 et +1, correspondant aux configurations où tous les sites contiennent respectivement un spin -1, un spin 0 et un spin +1, qui n'ont toutes les trois aucune interface. Ces configurations ne sont départagées que par leur magnétisation ; on observe que H(-1) > H(0) > H(+1).

L'étude de la compétition entre ces trois états s'est alors naturellement imposée comme un sujet majeur pour la métastabilité de ce modèle. Dans [START_REF] Emilio | Metastability and nucleation for the Blume-Capel model. Different mechanisms of transition[END_REF], Cirillo et Olivieri sont les premiers à travailler sur ce sujet, en s'intéressant notamment au cas où le tore est de taille fixe, et où le champ magnétique et le potentiel chimique sont faibles mais non-nuls. Comprenant que le système a une tendance à se diriger vers les états d'énergie faible lorsque la température tend vers 0, ils discutent des minima d'énergie locaux. On voit alors clairement que les configurations 0 et -1 sont des états métastables dans lequel le modèle reste longtemps avant de tenter de rejoindre +1. C'est pourquoi ils se sont intéressés à la trajectoire typique du modèle partant de -1 pour aller vers +1 ; au milieu des spins -1 -on parlera d'une mer de -1 -, un amas de spins 0, aussi appelé goutte, est créé. Cette goutte grandit progressivement, au prix d'une augmentation significative de l'énergie, jusqu'à atteindre une taille dite critique, à partir de laquelle elle continue de grandir, cette fois en diminuant l'énergie jusqu'à atteindre la configuration 0. La trajectoire de 0 à +1 est alors analogue avec une goutte de +1 qui grandit de façon similaire.

L'un des enjeux majeurs lorsque l'on étudie la métastabilité est l'estimation du temps de passage d'un état métastable à un autre. C'est donc une question qui a suscité, et suscite encore, beaucoup d'intérêt. Concernant le modèle de Blume-Capel, on peut notamment trouver dans les travaux de Manzo et Olivieri [START_REF] Manzo | Dynamical Blume-Capel model : competing metastable states at infinite volume[END_REF], réalisés dans le cas où le tore est de volume croissant et où potentiel chimique et champ magnétique sont strictement positifs, des premières majorations et minorations du temps d'atteinte de +1 partant de -1.

En 2013, Cirillo et Nardi [START_REF] Emilio | Relaxation height in energy landscapes : an application to multiple metastable states[END_REF] ont décrit de façon précise le paysage énergétique du modèle sur un tore de volume fixe, et ont surtout donné une meilleure estimation du temps d'atteinte de l'état +1 lorsque le modèle part de -1, présentant une erreur exponentielle : Dans cette thèse, nous montrons d'abord que, partant de -1, le processus atteint 0 avant d'atteindre +1, et nous caractérisons les configurations critiques. Comme mentionné ci-dessus, ces résultats ont été donné par Cirillo et Nardi. Néanmoins, nous présentons une nouvelle démonstration basée sur une inégalité fondamentale de la théorie du potentiel, faisant notamment le lien entre la probabilité d'atteindre 0 avant +1, et les capacités. De plus, les résultats précédents sur l'estimation des valeurs asymptotiques des temps d'atteinte s'appuient pour la plupart sur l'approche pathwise. Surtout, elles présentent des erreurs exponentielles ; nous étendons donc les résultats de Cirillo et Nardi en donnant des valeurs exactes. Nos travaux basés sur l'approche martingale de Beltrán et Landim présente donc deux avantages principaux : les estimations apparaissent comme plus fines, et la description du comportement métastable du modèle plus complète.

lim β→∞ 1 β ln E -1 [H +1 ] = Γ c , où H A = inf{t > 0 : σ t ∈
Cette thèse est constituée de deux parties. La première est constituée d'un article co-écrit avec Claudio Landim, et publié dans la Journal of Statistical Physics. La seconde est un article généralisant le premier au cas où le tore grandit au fur et à mesure que la température décroît.

Le chapitre 1 concerne le premier article. Dans celui-ci, nous étudions la métastabilité du modèle dans le cas où le tore est à taille fixée, principalement le comportement asymptotique du modèle. Le premier résultat principal concerne le passage de la configuration -1 à la configuration +1. Nous démontrons que la probabilité de ne pas passer par la configuration 0 tend vers 0 quand l'inverse de la température β tend vers l'infini :

lim β→∞ P β -1 [H +1 < H 0 ] = 0.
La démonstration de ce résultat illustre parfaitement l'efficacité de l'approche martingale de

lim β→∞ 1 e βΓc E -1 [H +1 ] = 1,
où e βΓc représente l'échelle de temps sur laquelle le modèle évolue lors du passage de -1 à +1, avec Γ c défini précédemment comme l'énergie permettant l'apparition d'une goutte critique. Là encore, la démonstration s'appuie essentiellement sur des résultats intermédiaires sur les capacités. Nous utilisons une identité dans [START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains[END_REF], exprimant cette espérance en fonction d'une capacité et d'une somme de probabilités, puis l'on démontre que la contribution de la majorité des termes de la somme est négligeable devant les termes correspondant aux trois configurations -1, 0 et +1, dont les puits sont les plus profonds du modèle. La réversibilité ainsi que quelques manipulations sur les capacités permettent alors d'obtenir cette limite. Dans de très nombreux systèmes observables dans la nature, l'évolution de la température influe sur la taille du domaine où évoluent les particules. Il apparaît donc comme pertinent de généraliser ces premiers résultats au cas où la taille du tore augmente quand la température décroît vers 0.

Dans le chapitre 2, nous nous intéressons cette fois-ci au cas où le tore est à taille croissante avec l'inverse de la température β. Si le cas où cette taille est fixe se ramène à l'estimation de capacités, celle-ci est rendue bien plus complexe par le fait que le nombre de chemins entre deux ensembles A et B soit désormais très grand, d'ordre exponentiel. Il s'agit alors de trouver des bornes fines pour ces capacités à l'aide des principes de Dirichlet et de Thomson.

Il découle de ces observations que la vitesse à laquelle la taille du tore augmente joue un rôle prépondérant. En effet, il est évident que le nombre total de configurations dans le modèle dépend de cette taille. Alors que, dans le cas où la taille du tore est fixe, il est suffisant de savoir que ce total est constant, il apparaît comme clair que cela ne l'est plus si la taille du tore dépend de β. Il est donc indispensable d'avoir une ou plusieurs conditions sur la vitesse de croissance. Plus précisément, nous supposons que la taille du tore croît exponentiellement rapidement :

lim β→∞ |Λ L | 1/2 e -[(n 0 +1)h-2]β + e -hβ et lim β→∞ |Λ L | 2 e -(2-h)β = 0 ,
où |Λ L | désigne le nombre de sites sur le tore. De plus amples explications sur ces conditions sont disponibles à la suite du Théorème 2.2.2. Une des principales difficultés réside encore dans l'estimations des capacités. En particulier, il faut être en mesure de compter le nombre de configurations d'un sousensemble donné. Lorsque la taille du tore est fixée, bien que le nombre total de configurations en dépende, celui-ci reste constant. Désormais, il est indispensable d'estimer le degré de dépendance en |Λ L | du nombre de configurations des sousensembles concernés par les capacités étudiées. A cette difficulté s'ajoute celle de l'estimation de l'énergie de ces mêmes configurations. En effet, la finesse de nos conditions exponentielles sur la vitesse de croissance de la taille du tore repose sur notre capacité à donner l'énergie minimale d'un sous-ensemble A ⊂ Ω L de configurations en fonction de la dépendance en |Λ L | du nombre total de configurations dans A.

Une première perspective possible pour ce travail est de tenter d'améliorer ces conditions. Nous donnons une première piste dans la Remarque 1.3.2 : il s'agirait de tenir compte des petites perturbations aléatoires pouvant se produire lorsqu'une goutte de spins 0 grandit. Ceci rendrait bien évidemment les estimations plus complexes.

Il serait également intéressant d'étudier la métastabilité dans d'autres cas concernant les valeurs de h et λ. Dans notre cas, nous supposions que λ = 0, ce qui avait pour effet de rendre symétriques les interactions de spins -1/0 et 0/+1. En particulier, la taille critique des gouttes de 0 est la même que celle des gouttes de +1. Si λ > 0, alors ces gouttes exhibent des tailles critiques différentes ; l'évolution d'une goutte de 0 est donc bien différente de celle d'une goutte de +1. Le comportement métastable s'en trouve alors bien plus difficile à étudier.

Chapitre 1

Metastability of the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field 1.1 Introduction

Since its first rigorous mathematical treatment [START_REF] Bovier | of Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Cassandro | Metastable behavior of stochastic dynamics : a pathwise approach[END_REF][START_REF] Freidlin | Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF][START_REF] Olivieri | Large deviations and metastability, volume 100 of Encyclopedia of Mathematics and its Applications[END_REF][START_REF] Penrose | Rigorous treatment of metastable states in the van der Waals-Maxwell theory[END_REF], metastability has been the subject of intensive investigation from different perspectives [START_REF] Barrera | Abrupt convergence and escape behavior for birth and death chains[END_REF][START_REF] Bianchi | Metastable states, quasistationary distributions and soft measures[END_REF][START_REF] Weinan | Towards a theory of transition paths[END_REF][START_REF] Fernandez | Conditioned, quasi-stationary, restricted measures and escape from metastable states[END_REF][START_REF] Metzner | Transition path theory for Markov jump processes[END_REF].

In [START_REF] Bovier | Metastability in stochastic dynamics of disordered mean-field models[END_REF][START_REF] Bovier | Metastability and low lying spectra in reversible Markov chains[END_REF] Bovier, Eckhoff, Gayrard and Klein, BEGK from now on, have shown that the potential theory of Markov chains can outperform large deviations arguments and provides sharp estimates for several quantities appearing in metastability, such as the expectation of the exit time from a well or the probability to hit a configuration before returning to the starting configuration.

Developing further BEGK's potential-theoretic approach, and with an intensive use of data reduction through trace processes, Beltrán and one of the authors of this paper, BL from now on, devised a scheme to describe the evolution of a Markov chain among the wells, particularly effective when the dynamics presents several valleys of the same depth [START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains[END_REF][START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains II, the nonreversible case[END_REF][START_REF] Beltrán | A martingale approach to metastability[END_REF]. The outcome of the method can be understood as a model reduction through coarse-graining, or as the derivation of the evolution of the slow variables of the chain.

In the case of finite state Markov chains [START_REF] Beltrán | Metastability of reversible finite state Markov processes[END_REF][START_REF] Landim | Metastability of finite state Markov chains : a recursive procedure to identify slow variables for model reduction[END_REF], under minimal assumptions, BL's method permits the identification of all slow variables, the derivation of the timescales at which they evolve and the characterization of their asymptotic dynamics.

In contrast with the pathwise approach [START_REF] Cassandro | Metastable behavior of stochastic dynamics : a pathwise approach[END_REF][START_REF] Olivieri | Large deviations and metastability, volume 100 of Encyclopedia of Mathematics and its Applications[END_REF] and the transition path theory [START_REF] Weinan | Towards a theory of transition paths[END_REF][START_REF] Metzner | Transition path theory for Markov jump processes[END_REF], BEGK's and BL's approach do not attempt to describe the tube of typical trajectories in a transition between two valleys, nor does it identify the critical configurations which are visited with high probability in such transitions.

Nevertheless, under weak hypotheses, introduced in Section 1.3 below, potentialtheoretic arguments together with data reduction through trace processes provide elementary identities and estimates which permit, without much effort, to characterize the critical configurations, and to compute the sub-exponential pre-factors of the expectation of hitting times. The purpose of this paper is to illustrate these assertions by examining the metastable behavior of the Blume-Capel model.

The Blume-Capel model is a two dimensional, nearest-neighbor spin system where the single spin variable takes three possible values : -1, 0 and +1. One can interpret it as a system of particles with spins. The value 0 of the spin at a lattice site corresponds to the absence of particles, whereas the values ±1 correspond to the presence of a particle with the respective spin.

The metastability of the Blume-Capel model has been investigated by Cirillo and Olivieri [START_REF] Emilio | Metastability and nucleation for the Blume-Capel model. Different mechanisms of transition[END_REF], Manzo and Olivieri [START_REF] Manzo | Dynamical Blume-Capel model : competing metastable states at infinite volume[END_REF], and more recently by Cirillo and Nardi [START_REF] Emilio | Relaxation height in energy landscapes : an application to multiple metastable states[END_REF], and Cirillo, Nardi and Spitoni [START_REF] Cirillo | Sum of exit times in series of metastable states in probabilistic cellular automata[END_REF]. We refer to [START_REF] Emilio | Metastability and nucleation for the Blume-Capel model. Different mechanisms of transition[END_REF][START_REF] Manzo | Dynamical Blume-Capel model : competing metastable states at infinite volume[END_REF] interest of the model and its role in the study of metastability.

We consider here a Blume-Capel model with zero chemical potential and a small positive magnetic field. We examine its metastable behavior in the zero-temperature limit in a large, but fixed, two-dimensional square with periodic boundary conditions. In this case, there are two metastable states, the configurations where all spins are equal to -1 or all spins equal to 0, and one ground state, the configuration where all spins are equal to +1.

The main results state that starting from -1, the configuration where all spins are equal to -1, the chain visits 0 before hitting +1. We also characterize the set of critical configurations. These results are not new and appeared in [START_REF] Emilio | Relaxation height in energy landscapes : an application to multiple metastable states[END_REF][START_REF] Emilio | Metastability and nucleation for the Blume-Capel model. Different mechanisms of transition[END_REF], but we present a proof which relies on a simple inequality from the potential theory of Markov chains. We compute the exact asymptotic values of the transition times, which corresponds to the life-time of the metastable states. The previous results on the transition time, based on the pathwise approach which relies on large deviations arguments, presented estimates with exponential errors. To complete the picture, we show that the expectation of the hitting time of the configuration 0 starting from -1 is much larger than the transition time. This phenomenon, which may seem contradicting the fact that the chain visits 0 before hitting +1, occurs because the main contribution to the expectation comes from the event that the chain first hits +1 and then visits 0. The very small probability of this event is compensated by the very long time the chain remains at +1. Finally, we prove the metastable behavior of the Blume-Capel model in the sense of BL. Let Σ be the set of configurations and let V -1 , V 0 , V +1 be neighborhoods of the configurations -1, 0, +1, respectively. For instance, V -1 = {-1}, V 0 = {0}, V +1 = {+1}. Denote the inverse of the temperature by β, and let φ : Σ → {-1, 0, 1, β} be the projection defined by

φ(σ) = a=-1,0,+1 a 1{σ ∈ V a } + β 1 σ ∈ a=-1,0,+1 V a .
We prove that there exists a time scale θ β for which φ(σ(tθ β )) converges, as β → ∞, to a Markov chain in {-1, 0, +1}. The point +1 is an absorbing point for this Markov chain, and the other jump rates are given by r(-1, 0) = r(0, 1) = 1 , r(-1, 1) = r(0, -1) = 0 .

As we said above this result can be interpreted as a model reduction by coarsegraining, or as the identification of a slow variable, φ, whose evolution is asymptotically Markovian.

The article is divided as follows. In Section 1.2, we state the main results. In Section 1.3, we introduce the main tools used throughout the article and we present general results on finite-state reversible Markov chains. In section 1.4, we examine the transition from -1 to 0, and in Section 1.5 the one from 0 to +1. In Section 1.6, we analyze the hitting time of 0 starting from -1. In the last section, we prove the metastable behavior of the Blume-Capel model with zero chemical potential as the temperature vanishes.

Notation and main results

Fix L > 1 and let Λ L = T L × T L , where T L = {1, • • • , L} is the discrete, onedimensional torus of length L. Denote the configuration space by Ω = {-1, 0, 1} Λ L , and by the Greek letters σ, η, ξ the configurations of Ω. Hence, σ(x), x ∈ Λ L , represents the spin at x of the configuration σ.

Fix an external field 0 < h < 1, and denote by H : Ω → R the Hamiltonian given by

H(σ) = σ(y) -σ(x) 2 -h x∈Λ L σ(x) , (1.1)
where the first sum is carried over all unordered pairs of nearest-neighbor sites of Λ L . Let n 0 = [2/h], where [a] represents the integer part of a ∈ R + . We assume that L > n 0 + 3. Denote by β > 0 the inverse of the temperature and by µ β the Gibbs measure associated to the Hamiltonian H at inverse temperature β,

µ β (σ) = 1 Z β e -βH(σ) , (1.2)
where Z β is the partition function, the normalization constant which turns µ β a probability measure. Denote by -1, 0, +1 the configurations of Ω with all spins equal to -1, 0, +1, respectively. These three configurations are local minima of the energy H, H(+1) < H(0) < H(-1), and +1 is the unique ground state.

The Blume-Capel dynamics is the continuous-time Markov chain on Ω, denoted by {σ t : t ≥ 0}, whose infinitesimal generator L β acts on functions f : Ω → R as

(L β f )(σ) = x∈Λ L R β (σ, σ x,+ ) f (σ x,+ ) -f (σ) + x∈Λ L R β (σ, σ x,-) f (σ x,-) -f (σ) . (1.3)
In this formula, σ x,± represents the configuration obtained from σ by modifying the spin at x as follows,

σ x,± (z) := σ(x) ± 1 mod 3 if z = x σ(z) if z = x
and the rates R β are given by

R β (σ, σ x,± ) = exp -β H(σ x,± ) -H(σ) + , x ∈ Λ L ,
where a + , a ∈ R, stands for the positive part of a : a + = max{a, 0}.

Clearly, the Gibbs measure µ β satisfies the detailed balance condition

µ β (σ)R β (σ, σ x,± ) = min µ β (σ) , µ β (σ x,± ) = µ β (σ x,± )R β (σ x,± , σ) ,
and is therefore reversible for the dynamics. Denote by D(R + , Ω) the space of right-continuous functions x : R + → Ω with left-limits and by P σ = P β σ , σ ∈ Ω, the probability measure on the path space D(R + , Ω) induced by the Markov chain σ t starting from σ. Expectation with respect to P σ is represented by E σ .

Denote by H A , H + A , A ⊂ Ω, the hitting time and the time of the first return to A, respectively :

H A = inf t > 0 : σ t ∈ A , H + A = inf t > τ 1 : σ t ∈ A , ( 1
P -1 [H +1 < H 0 ] = 0 .
Recall the definition of n 0 introduced just below (1.1). Denote by R l the set of configurations in {-1, 0, +1} Λ L for which (a) There is a n 0 × (n 0 + 1)-rectangle or a (n 0 + 1) × n 0 -rectangle of 0-spins ; The set of configurations R l 0 is defined similarly, replacing 0-spins by +1-spins and -1-spins by 0-spins. In particular, configurations in R l 0 do not have -1 spins and have n 0 (n 0 + 1) + 1 spins equal to +1.

The next result states that, starting from -1, the chain reaches the set R l before hitting 0. Proposition 1.2.2. We have that

lim β→∞ P -1 [H R l < H 0 ] = 1 , lim β→∞ P 0 [H R l 0 < H +1 ] = 1 .
Denote by λ β (σ), σ ∈ Ω, the holding rates of the Markov chain σ t , and by p β (η, ξ), η, ξ ∈ Ω, the jump probabilities, so that R β (η, ξ) = λ β (η)p β (η, ξ). Let M β (η) = µ β (η)λ β (η) be the stationary measure for the embedded discrete-time Markov chain. The index β of M β will be frequently omitted below.

Denote by cap(A, B) the capacity between two disjoint subsets A, B of Ω :

cap(A, B) = cap β (A, B) = σ∈A M β (σ) P σ [H B < H + A ] , (1.5)
and let

θ β = µ β (-1) cap(-1, {0, +1}) • (1.6)
Lemma 4.2 in [START_REF] Beltrán | Metastability of reversible finite state Markov processes[END_REF] states that cap(-1, {0, +1})/ exp{-βH} converges to a constant as β ↑ ∞, where H is the energy of the saddle configuration separating -1 from the set {0, +1}. In particular, H -H(-1) represents the energy barrier between -1 and {0, +1} and θ β the time-scale at which the Blume-Capel model reaches the set {0, +1} starting from the local minima -1. In this sense, Proposition 1.2.3 below asserts that R l corresponds to the set of saddle or critical configurations between -1 and {0, +1}, while R l 0 the ones between 0 and {-1, +1}. Since we expect, by symmetric considerations, to reach +1 before -1 when starting from 0 (cf. Assertion 1.5.3), θ β represents the time-scale in which the Blume-Capel model reaches the ground state +1 starting from the local minima -1 or 0. Proposition 1.2.3. For any configuration η ∈ R l and any configuration ξ ∈ R l 0 ,

lim β→∞ cap(-1, {0, +1}) µ β (η) = 4(2n 0 + 1) 3 |Λ L | = lim β→∞ cap(0, {-1, +1}) µ β (ξ) .
The first identity of this proposition is proved in Section 1.4 and the second one in Section 1.5. Proposition 1.2.4. The expected time to visit the ground state starting from -1 and from 0 are given by

lim β→∞ 1 θ β E -1 [H +1 ] = 2 , lim β→∞ 1 θ β E 0 [H +1 ] = 1 .
We have seen in Proposition 1.2.1 that starting from -1 the process reaches 0 before visiting +1. In contrast, the next proposition shows that the main contribution to the expectation E -1 [H 0 ] comes from the event in which the process, starting from -1, first visits +1, remains there for a very long time and then reaches 0. Proposition 1.2.5. We have that

1 θ β E -1 [H 0 ] = 1 + o(1) µ β (+1) µ β (0) P -1 [H +1 < H 0 ] ,
where o( 1) is an expression which vanishes as β ↑ ∞, and (1.7)

lim β→∞ µ β (+1) µ β (0) P -1 [H +1 < H 0 ] = ∞ . A self-avoiding path γ from A to B, A, B ⊂ Ω, A ∩ B = ∅, is a sequence of configurations (ξ 0 , ξ 1 , . . . , ξ n ) such that ξ 0 ∈ A, ξ n ∈ B, ξ j ∈ A ∪ B, 0 < j < n, ξ i = ξ j , i = j, R β (ξ i , ξ i+1 ) > 0, 0 ≤ i < n.
Thus, H(γ) represents the highest energy level attained by the path γ, while H(A, B) denotes the energy of the saddle point between the sets A and B.

Let M = {-1, 0, +1} be the set of ground configurations of the main wells, and let V η , η ∈ M, be a neighborhood of the configuration η. We assume that all configurations σ ∈ V η , σ = η, fulfill the conditions

H(σ) > H(η) , H(η, σ) -H(η) < H(-1, {0, +1}) -H(-1) .
(1.8)

The right hand side in the second condition represents the energetic barrier the chain needs to surmount to reach the set {0, +1} starting from -1, while the left hand side represents the energetic barrier to go from η to σ. It follows from Proposition 1.2.3 that

H(-1, {0, +1}) -H(-1) = H(0, {-1, +1}) -H(0) . (1.9)
Indeed, as we pointed out below (1.6), the set R l corresponds to the set of saddle points between -1 and {0, +1} so that µ β (η) = exp{-β H(-1, {0, +1})} for η ∈ R l . The same observation is in force for the set R l 0 so that

µ β (ξ) = exp{-β H(0, {-1, +1})}
for ξ ∈ R l 0 . These two equations permit to compute the energies H(-1, {0, +1}), H(0, {-1, +1}) and to check the identy (1.9).

We may therefore replace the expression on the right hand side of (1.8) by the one on the right hand side of the previous formula.

Clearly, V η = {η}, η ∈ M, is an example of neighborhoods satisfying (1.8). Let V be the union of the three neighborhoods, V = ∪ η∈M V η , and let π : M → {-1, 0, 1} be the application which provides the magnetization of the states -1, 0, +1 :

π(-1) = -1, π(0) = 0, π(+1) = 1. Denote by Ψ = Ψ V : Ω → {-1, 0, 1, [β]} the projection defined by Ψ(σ) = π(η) if σ ∈ V η , Ψ(σ) = [β], otherwise : Ψ(σ) = η∈M π(η) 1{σ ∈ V η } + [β] 1 σ ∈ η∈M V η .
Recall from [START_REF] Landim | A topology for limits of Markov chains[END_REF] the definition of the soft topology. Theorem 1.2.6. The speeded-up, hidden Markov chain X β (t) = Ψ σ(θ β t) converges in the soft topology to the continuous-time Markov chain X(t) on {-1, 0, 1} in which 1 is an absorbing state, and whose jump rates are given by r(-1, 0) = r(0, 1) = 1 , r(-1, 1) = r(0, -1) = 0 . Remark 1.2.7. Denote by B η , η ∈ M, the basin of attraction of η :

B η = {σ : lim β→∞ P σ [H M\{η} < H η ] = 0} .
We prove in (1.48) that V η ⊂ B η .

Metastability of reversible Markov chains

We present in this section some results on reversible Markov chains. Consider two nonnegative sequences (a N : N ≥ 1), (b N : N ≥ 1). The notation a N ≺ b N (resp.

a N b N ) indicates that lim sup N →∞ a N /b N = 0 (resp. lim sup N →∞ a N /b N < ∞), while a N ≈ b N means that a N b N and b N a N .
A set of nonnegative sequences (a r N : N ≥ 1), r ∈ R, is said to be ordered if for all r = s ∈ R arctan(a r N /a s N ) converges. Fix a finite set E. Consider a sequence of continuous-time, E-valued Markov chains {η N t : t ≥ 0}, N ≥ 1. We assume, throughout this section, that the chain η N t is irreducible, that the unique stationary state, denoted by µ N , is reversible, and that the jump rates of the chain η N t , denoted by R N (x, y), x = y ∈ E, satisfy the following hypothesis. Let Z + = {0, 1, 2, . . .}, let B be the bonds of E : B = {(x, y) ∈ E × E : x = y ∈ E}, and let A m , m ≥ 1, be the set of functions k : B → Z + such that (x,y)∈B k(x, y) = m. Assumption 1.3.1. We assume that for every m ≥ 1 the set of sequences

(x,y)∈B R N (x, y) k(x,y) : N ≥ 1 , k ∈ A m is ordered.
This assumption is slightly weaker than the hypotheses (2.1), (2.2) in [START_REF] Beltrán | Metastability of reversible finite state Markov processes[END_REF], but strong enough to derive all results presented in that article [START_REF] Landim | Metastability of finite state Markov chains : a recursive procedure to identify slow variables for model reduction[END_REF]. It is also not difficult to check that the Blume-Capel model introduced in the previous section fulfills Assumption 1.3.1.

We adopt here similar notation to the one introduced in the previous section. For example, λ N (x) represents the holding rates of the Markov chain η N t , p N (x, y), x, y ∈ E, the jump probabilities, and M N (x) = µ N (x)λ N (x) a stationary measure of the embedded discrete-time Markov chain. Analogously, P x = P N x , x ∈ E, represents the distribution of the Markov chain η N t starting from x and E x the expectation with respect to P x .

Denote by H A (resp. H + A ), A ⊂ E, the hitting time of (resp. the return time to) the set A, introduced in (1.4), and by cap(A, B) the capacity between two disjoint subsets A, B of E, as defined in (1.5).

Identities (1.10)-(1.12) below are well known and will be used often (cf. [START_REF] Bovier | of Grundlehren der Mathematischen Wissenschaften[END_REF]Lemma 8.4]). Let A, B be two disjoint subsets of E and let x be a point which does not belong to A ∪ B. We claim that

P x [H A < H B ] = P x [H A < H + B∪{x} ] P x [H A∪B < H + x ] • (1.10)
To prove this identity, intersect the event {H A < H B } with the set {H + x < H A∪B } and its complement, and then apply the strong Markov property to get that

P x [H A < H B ] = P x [H + x < H A∪B ] P x [H A < H B ] + P x [H A < H + B∪{x} ] .
To obtain (1.10) it remains to subtract the first term on the right hand side from the left hand side.

Multiply and divide the right hand side of (1.10) by M (x) and recall the definition of the capacity to obtain that

P x [H A < H B ] = M (x)P x [H A < H + B∪{x} ] cap(x, A ∪ B) ≤ M (x)P x [H A < H + x ] cap(x, A ∪ B) • (1.11)
Hence, by definition of capacity and since, by [17, Lemma 2.2], the capacity is monotone.

P x [H A < H B ] ≤ cap(x, A) cap(x, A ∪ B) ≤ cap(x, A) cap(x, B) • (1.
cap(A, B) = cap(B, A) = y∈B M (y)P y [H A < H + B ] ≤ x∈A y∈B M (y)P y [H x < H + B ] . Therefore, since cap(B, x) = cap(x, B), cap(A, B) ≤ x∈A cap(x, B) ≤ |A| max x∈A cap(x, B) , (1.14)
where |A| stands for the cardinality of A. Repeating this argument for B in place of A, we conclude the proof of (1.13).

Let y) and note that G N is symmetric. In the electrical network interpretation of reversible Markov chains, G N (x, y) represents the conductance of the bond (x, y). Recall that a self-avoiding path

G N : E × E → R + be given by G N (x, y) = µ N (x)R N (x,
γ from A to B, A, B ⊂ E, A ∩ B = ∅, is a sequence of sites (x 0 , x 1 , . . . , x n ) such that x 0 ∈ A, x n ∈ B, x j ∈ A ∪ B, 0 < j < n, x i = x j , i = j, R N (x i , x i+1 ) > 0, 0 ≤ i < n.
Denote by Γ A,B the set of self-avoiding paths from A to B and let

G N (A, B) := max γ∈Γ A,B G N (γ) , G N (γ) := min 0≤i<n G N (x i , x i+1 ) .
By [START_REF] Beltrán | Metastability of reversible finite state Markov processes[END_REF]Lemma 4.2], for every disjoint subsets A, B of E, the limit lim

N →∞ cap(A, B) G N (A, B)
exists and belongs to (0, ∞) .

(1.15)

Remark 1.3.2. Suppose that the stationary state µ N is a Gibbs measure associated to an energy H, and that we are interested in the Metropolis dynamics :

µ N (x) = Z -1 N exp{-N H(x)}, where Z N is the partition function, R N (x, y) = exp{-N [H(y) - H(x)] + }. In this context, G N (x, y) = Z -1 N exp{-N max[H(x), H(y)]}. In particular, for a path γ = (x 0 , x 1 , . . . , x n ), G N (γ) = Z -1 N exp{-N max i H(x i )}, and for two disjoint subsets A, B of E, G N (A, B) = 1 Z N exp -N min γ∈Γ A,B max x∈γ H(x) = 1 Z N exp -N H(x A,B ) = µ N (x A,B ) ,
where x A,B represents the configuration with highest energy in the optimal path joining A to B.

Lemma 1.3.3. Let E 1 be a subset of E. Assume that for every y ∈ E 1 , z ∈ E 1 such that µ(z) µ(y), cap(y, z) µ(z) ≺ cap(y, E 1 ) µ(y) • (1.16)
Then, for any Section 6] for the definition of trace), and let q(y, z)

B ⊂ E 1 , x ∈ E 1 \ B, E x [H B ] = 1 + o(1) 1 cap(x, B) y∈E 1 µ(y) P y [H x < H B ] . Proof. By [2, Proposition 6.10], E x [H B ] = 1 cap(x, B) y∈E µ(y) P y [H x < H B ] . Denote by P 1 z , z ∈ E 1 , the distribution of the trace of σ(t) on the set E 1 starting from z (cf. [2,
= P y [H E 1 = H z ], y ∈ E 1 , z ∈ E 1 .
Decomposing the previous sum according to y ∈ E 1 , y ∈ E 1 , since B and x are contained in E 1 , we can write it as

y∈E 1 µ(y) P 1 y [H x < H B ] + y ∈E 1 z∈E 1 µ(y) q(y, z) P 1 z [H x < H B ] = y∈E 1 µ(y) P 1 y [H x < H B ] + z∈E 1 P 1 z [H x < H B ] y ∈E 1 µ(y) q(y, z) .
(1.17)

We claim that for

y ∈ E 1 , z ∈ E 1 , µ(y) q(y, z) = µ(y) P y [H z < H E 1 \{z} ] ≺ µ(z) . ( 1.18)
If µ(y) ≺ µ(z), there is nothing to prove. Assume that µ(z) µ(y). In this case, by (1.12) and by (1.16), the second term in the previous expression is bounded by

µ(y) cap(y, z) cap(y, E 1 ) ≺ µ(z) .
This proves claim (1.18) and that the second term in the last equation of (1.17) is of smaller order than the first, as asserted.

Remark 1.3.4. In Lemma 1.3.3, the set E 1 has to be interpreted as the union of wells. In the set-up of the Metropolis dynamics introduced in Remark 1.3.2, by (1.15) and Remark 1.3.2, for two disjoint subsets

A, B of E, cap(A, B)/µ N (x A,B ) converges, as N ↑ ∞, to a real number in (0, ∞). Hence, assumption (1.16) requires that for all z ∈ E 1 , y ∈ E 1 such that H(y) ≤ H(z), H(x y,E 1 ) -H(y) < H(x y,z ) -H(z) . (1.19)
In other words, it requires the energy barrier from y to E 1 to be smaller than the one from z to y.

The condition (1.19) may seem unnatural, as one would expect on the right hand side H(x z, Ȇz ) -H(z) instead of H(x y,z ) -H(z), where Ȇz represents the union of the wells which do not contain z. However, since in the applications the set E 1 represents the union of wells, and since H(y) ≤ H(z), to reach y from z the chain has to jump from one well to another and therefore one should have

H(x z, Ȇz ) -H(z) ≤ H(x y,z ) -H(z). Lemma 1.3.5. Fix two points a = b ∈ E. The set of sequences µ N (x)P x [H a < H b ], x ∈ E \ {b}, is ordered. Proof. Fix two points x = y ∈ E \{b}. We need to show that the ratio µ N (x)P x [H a < H b ]/µ N (y)P y [H a < H b ] either converges to some value in [0, ∞), or increases to ∞.
Assume that x = a, y = a, and consider the trace of the process η N (t) on A = {a, b, x, y}. By [2, Section 6], the stationary measure of the trace is the measure µ N conditioned to A. Denote by P A z the distribution of the trace starting from z. It is clear that

P z [H a < H b ] = P A z [H a < H b ]. Therefore, µ N (x)P x [H a < H b ] µ N (y)P y [H a < H b ] = µ A N (x)P A x [H a < H b ] µ A N (y)P A y [H a < H b ]
,

where µ A N represents the measure µ N conditioned to A. Since A has only four elements, it is not difficult to show that

P A x [H a < H b ] = p A (x, a) + p A (x, y)p A (y, a) 1 -p A (x, y)p A (y, x) ,
where p A (z, z ) represents the jump probabilities of the trace process. In particular, multiplying the numerator and the denominator of the penultimate ratio by λ A (x)λ A (y), where λ A stands for the holding rates of the trace process, yields that the penultimate ratio is equal to

µ A N (x){λ A (y)R A (x, a) + R A (x, y)R A (y, a)} µ A N (y){λ A (x)R A (y, a) + R A (y, x)R A (x, a)}
, where R A is the jump rates of the trace process. By reversibility of the trace process, this expression is equal to

λ A (y)R A (a, x) + R A (y, x)R A (a, y) λ A (x)R A (a, y) + R A (x, y)R A (a, x) • By [3, Lemma 4.3], the set of jump rates R A (x, y) satisfies Assumption 1.3.1. Since λ A (z) = z ∈A,z =z R A (z, z )
, by Assumption 1.3.1, the previous expression either converges to some a ∈ [0, ∞), or increases to +∞. This completes the proof of the assertion in the case where x, y ∈ {a, b}.

The case where x = a or y = a is simpler and left to the reader.

Proofs of Propositions 1.2.1, 1.2.2 and 1.2.3

We examine in this section the metastable behavior of the Blume-Capel model starting from -1. We first consider isovolumetric inequalities. Denote by

• the Euclidean norm of R 2 . A subset A of Z 2 is said to be connected if for every x, y ∈ A, there exists a path γ = (x = x 0 , x 1 , . . . , x n = y) such that x i ∈ A, x i+1 -x i = 1, 0 ≤ i < n. Denote by C n , n ≥ 1,
the class of connected subset of Z 2 with n points and by P (A) the perimeter of a set A ∈ C n :

P (A) = #{(x, y) ∈ Z 2 : x ∈ A , y ∈ A , x -y = 1} .
where #B stands for the cardinality of B.

Assertion 1.4.1. For every

A ∈ C n , n ≥ 1, P (A) ≥ 4 √ n.
Proof. For A ∈ C n , denote by R the smallest rectangle which contains A, and by a ≤ b the length of the sides of the rectangle R. Since A is connected, and since R is the smallest rectangle which contains

A, P (A) ≥ 2(a + b) ≥ 2 min{a + b : a, b ∈ R , ab ≥ n} = 4 √ n. Assertion 1.4.2. A set A ∈ C m , m = n 0 (n 0 + 1)
, is either a n 0 × (n 0 + 1) rectangle or has perimeter P (A) ≥ 4(n 0 + 1).

Proof. Fix A ∈ C m , and recall the notation introduced in the proof of the previous assertion. We may restrict our study to the case where the length of the shortest side of R, denoted by a, is less than or equal to n 0 , otherwise the perimeter is larger than or equal to 4(n 0 + 1). If a = n 0 , either b = n 0 + 1, in which case, to match the volume, A must be a n 0 × (n 0 + 1) rectangle, or b ≥ n 0 + 2, in which case the the perimeter is larger than or equal to 4(n 0 + 1). If a = n 0 -j for some j ≥ 1, then b = n 0 + k for some k ≥ 1 because the volume has to be at least n 2 0 . Actually, we need (n 0 + k)(n 0 -j) ≥ n 0 (n 0 + 1), i.e., (k -j)n 0 ≥ n 0 + kj. This forces k -j ≥ 2 and, in consequence, the perimeter P ≥ 4(n 0 + 1).

We may extend the definition of the energy H introduced in (1.1) to configuration in {-1, 0, 1} Z 2 . For such configurations, while H(σ) is not well defined, H(σ)-H(-1) is well defined if σ x = -1 for all but a finite number of sites.

Denote by 2 , and let σ be a configuration of {-1, 0, 1} Z 2 whose spins in A are equal to +1 and whose spins in ∂ + A are either 0 or -1. Let σ b ?e the configuration obtained from σ by switching all spins in A to 0. Then, H(σ) ≥ H(σ ) + 2.

∂ + A the outer boundary of a connected finite subset A of Z 2 : ∂ + A = {x ∈ A : ∃ y ∈ A s.t. y -x = 1}. Assertion 1.4.3. Let A ∈ C n , 1 ≤ n ≤ (n 0 + 1)
Proof. By definition of the energy and since A has n points, H(σ) -H(σ ) = -hn + P 0 +3P -1 ≥ -hn+P , where P 0 (resp. P -1 ) represents the number of unordered pairs {x, y} such that x ∈ A, y ∈ ∂ + A, σ y = 0 (resp. σ y = -1), and where P = P 0 + P -1 is the perimeter of the set A.

It remains to show that P -hn ≥ 2. For 1 ≤ n ≤ 3, this follows by inspecting all cases, keeping in mind that h < 1 ≤ 2. Next, assume that n ≥ 4. By hypothesis, and since

n 0 ≥ 2, (2/3) √ n ≤ (2/3)(n 0 + 1) ≤ n 0 < 2/h so that hn < 3 √ n. Hence, by Assertion 1.4.1, hn < 4 √ n - √ n ≤ P -2. Let A(σ) = {x ∈ Z 2 : σ x = -1}, σ ∈ {-1, 0, 1} Z 2 .
Denote by B the boundary of the valley of -1 formed by the set of configurations with n 0 (n 0 + 1) sites with spins different from -1 :

B = σ ∈ {-1, 0, 1} Z 2 : |A(σ)| = n 0 (n 0 + 1) .
Sometimes, we consider B as a subset of Ω. Denote by R the subset of B given by

R = σ ∈ {-1, 0} Z 2 : A(σ) is a n 0 × (n 0 + 1) rectangle .
Note that the spins of a configuration σ ∈ R are either -1 or 0 and that all configurations in R have the same energy.

Assertion 1.4.4. We have that

H(σ) ≥ H(ζ) + 2 for all σ ∈ B \ R, ζ ∈ R.
Proof. Fix a configuration σ ∈ B. Let σ * be the configuration of {-1, 0} Z 2 obtained from σ by switching all +1 spins to 0. Applying Assertion 1.4.3 k times, where k is the number of connected components formed by +1 spins, we obtain that H(σ) ≥ H(σ * ) + 2k. It is therefore enough to prove the lemma for configurations

σ ∈ {-1, 0} Z 2 .
Let σ be a configuration in B∩{-1, 0} Z 2 . If A(σ) is not a connected set, by gluing the connected components of A(σ), we reach a new configuration σ * ∈ B∩{-1, 0} Z 2 such that A(σ * ) ∈ C m , m = n 0 (n 0 + 1). Since by gluing two components, the volume remains unchanged, but the perimeter decreases at least by 2, H(σ) ≥ H(σ * ) + 2. It is therefore enough to prove the lemma for those configuration in Denote by R + the set of configurations in {-1, 0, +1} Z 2 in which there are n 0 (n 0 + 1)+1 spins which are not equal to -1. Of these spins, n 0 (n 0 +1) form a n 0 ×(n 0 +1)rectangle of 0 spins. The remaining spin not equal to -1 is either 0 or +1.

B ∩ {-1, 0} Z 2 for which A(σ) is a connected set. Finally, fix a configuration in B ∩ {-1, 0} Z 2 for which A(σ) is a connected set different from a n 0 × (n 0 + 1)
It is clear that starting from -1 the set (B \ R) ∪ R + is hit before the chain attains the set {0, +1} :

H B + < H {0,+1} P -1 a.s. , where B + = (B \ R) ∪ R + .
(

Let R a ⊂ R + be the set of configurations for which the remaining spin is a 0 spin attached to one of the sides of the rectangle. Note that all configurations of R a have the same energy and that

H(ξ) = H(ζ) + 2 -h if ξ ∈ R a , ζ ∈ R. In particular, H(σ) ≥ H(ξ) + h , σ ∈ B \ R , ξ ∈ R a . (1.21)
On the other hand, for a configuration

η ∈ R + \ R a , H(η) ≥ H(ζ) + 4 -h if ζ ∈ R, so that H(η) ≥ H(ξ) + 2 , η ∈ R + \ R a , ξ ∈ R a . (1.22)
Recall the notation introduced just above Remark 1.3.2. For two disjoint subsets A to B of Ω, denote by ξ A,B , the configuration with highest energy in the optimal path joining A to B. By Remark 1.3.2 and (1.15), the limit

lim β→∞ cap(A, B) µ β (ξ A,B )
exists and takes value in (0, ∞) .

(1.23)

In particular, for every subset B of Ω and every configuration σ ∈ B,

cap(σ, B) 1 Z β e -βH(σ) . (1.24) Assertion 1.4.5. For all configurations ξ ∈ R a , cap(ξ, -1) ≈ Z -1 β exp{-βH(ξ)}. Proof. Fix ξ ∈ R a . By (1.23), (1.24), it is enough to exhibit a path γ = (ξ = ξ 0 , ξ 1 , . . . , ξ n = -1) from ξ to -1 such that max i H(ξ i ) = H(ξ).
Consider the path γ = (ξ = ξ 0 , ξ 1 , . . . , ξ n = -1), n = n 0 (n 0 + 1) + 1, constructed as follows. ξ 1 is the configuration obtained from ξ by switching the attached spin from 0 to -1. Clearly, H(ξ 1 ) = H(ξ) -2 + h, and ξ 1 consists of a n 0 × (n 0 + 1) rectangle of 0 spins.

The portion (ξ 1 , . . . , ξ n 0 +1 ) of the path γ is constructed by flipping, successively, from 0 to -1, all spins of one of the shortest sides of the rectangle, keeping until the last step the perimeter of the set A(ξ i ) equal to 4n 0 (n 0 + 1). In particular, ξ n 0 +1 consists of a n 0 × n 0 square of 0 spins, H(ξ i+1 ) = H(ξ i ) + h for 1 ≤ i < n 0 , and

H(ξ n 0 +1 ) = H(ξ n 0 ) -2 + h.
The energy of this piece of the path attains its maximum at ξ n 0 and H(ξ

n 0 ) = H(ξ 1 ) + (n 0 -1)h = H(ξ) -2 + n 0 h < H(ξ).
The path proceed in this way by always flipping from 0 to -1 all spins of one of the shortest sides. It is easy to check that H(ξ i ) < H(ξ) for all 1 ≤ i ≤ n, proving the assertion.

Assertion 1.4.6. For every σ ∈ B + \ R a , lim β→∞ P -1 [H σ = H B + ] = 0.
Proof. Fix σ ∈ B + \ R a and ξ ∈ R a . By (1.12), by the monotonicity of the capacity, stated in [17, Lemma 2.2], by (1.24), and by Assertion 1.4.5,

P -1 [H σ = H B + ] ≤ cap(σ, -1) cap(B + , -1) ≤ C 0 Z β e -βH(σ) cap(ξ, -1) ≤ C 0 e -β{H(σ)-H(ξ)}
for some finite constant C 0 independent of β. By (1.21), (1.22), this expression vanishes as β ↑ ∞, proving the assertion.

Denote by S the set of stable configurations :

S = σ ∈ Ω : lim β→∞ λ β (σ) = 0 . (1.25)
The stable configurations are formed as follows. On a background of negative spins, place vacant rectangles of length and width larger than or equal to 2, and vacant rings of length or width larger than 2. Note that inserting a vacant ring of size L × L is not excluded. We just require the graph distance between vacant rectangles and vacant rings to be greater than or equal to 3, and we do not allow the coexistence of horizontal and vertical vacant rings. Inside the vacant rectangles and the vacant rings, embed positive rectangles along the same rules. This means that the length and the width of the positive rectangles should be larger than or equal to 2, that the length or the width of the positive rings should be larger than or equal to 2, that the graph distance between positive rectangles and positive rings should be greater than or equal to 3. Note that the previous rules do not allow the coexistence of horizontal and vertical positive rings. Furthermore, we do not allow positive spins to have negative spins as neighbors, a layer of vacant sites should separate positive from negative spins.

The next assertion is the only one in which capacities are not used to derive the needed bounds, because we estimate the probability of reaching a state which can be attained through paths in which the energy never increase. The argument, though, is fairly simple. Denote by R c , R i the configurations of R a in which the extra particle is attached to the corner, interior of the rectangle, respectively (cf. Figure 1.1).

Assertion 1.4.7. For σ ∈ R c and σ ∈ R i , lim β→∞ P σ [H σ + = H S ] = 1/2 and lim β→∞ P σ [H σ -= H S ] = 1/2 , lim β→∞ P σ [H σ + = H S ] = 2/3 and lim β→∞ P σ [H σ -= H S ] = 1/3 ,
where σ -is the configuration obtained from σ or σ by flipping to -1 the attached 0 spin, and σ + is the configuration whose set A(σ + ), formed only by 0 spins, is the smallest rectangle which contains A(σ).

Proof. Suppose that the extra 0 spin is not attached to the corner of the rectangle. Denote by σ 1 , σ 2 the configurations obtained from σ by flipping from -1 to 0 one of the two -1 spins which has two neighbor spins equal to 0, and let 1) for all the other configurations, where o(1) represents an expression which vanishes as β ↑ ∞. This shows that p β (σ , σ j ) converges to 1/3, 0 ≤ j ≤ 2. We may repeat this argument to show that from σ j , j = 1, 2, one reaches S at σ + with a probability asymptotically equal to 1. The argument is similar if the extra spin is attached to the corner. Assertion 1.4.8. Fix a configuration σ ∈ S for which A(σ) is a m × n rectangle of 0 spins in a sea of -1 spins. Assume that m ≤ n. Then,

σ 0 = σ -. By definition, R β (σ , σ j ) = 1, 0 ≤ j ≤ 2, and R β (σ , σ ) = o(
lim β→∞ P σ [H B = H S\{σ} ] = 1 .
In this equation, if n 0 < m, n ≤ L -3, B is the set of four configurations in which a row or a column of 0 spins is added to the rectangle A(σ). If n 0 < m < n = L-2, the set B is a triple which includes a band of 0 spins of width m and two configurations in which a row or a column of 0 spins of length n is added to the rectangle A(σ).

If n 0 < m ≤ L -3, n = L, the set B is a pair formed by two bands of 0 spins of width m + 1. If n 0 < m = n = L -2, B is a pair of two bands of width L -2. If n 0 < m = L -2, n = L, B = {0}. Finally, if 2 ≤ m ≤ n 0 , n ≥ 3, the set B is the pair (quaternion if m = n) of
configurations in which a row or a column of 0 spins of length m is removed from the rectangle A(σ), and if m = n = 2, B = {-1}. The case m = 1 is not considered because such a configuration is not stable.

Proof. The assertion follows from inequality (1.12), and from estimates of cap(σ, S \ {σ}), cap(σ, S\[B∪{σ}]). In the case m > n 0 , cap(σ, S\{σ})

≈ µ β (σ) exp{-β(2-h)} and cap(σ, S\[B∪{σ}]) ≺ µ β (σ) exp{-β(2-h)}, while in the case m ≤ n 0 , cap(σ, S\ {σ}) ≈ µ β (σ) exp{-β(m -1)h} and cap(σ, S \ [B ∪ {σ}]) ≺ µ β (σ) exp{-β(m -1)h}.
Lemma 1.4.9. For every σ ∈ B + ,

lim β→∞ P -1 [H σ = H B + ] = 1 |R a | 1{σ ∈ R a } ,
where |R a | represents the number of configurations in R a .

Proof. In view of Assertion 1.4.6, we may restrict our attention to σ ∈ R a . Fix a reference configuration σ * in R a . By (1.10) and by definition of the capacity,

P -1 [H σ = H B + ] = M (-1) P -1 [H σ = H + B + ∪{-1} ] cap(-1, B + ) •
By reversibility, the numerator of this expression is equal to

M (σ) P σ [H -1 = H + B + ∪{-1} ] = µ β (σ) λ(σ) P σ [H -1 = H + B + ∪{-1} ] .
By Assertions 1.4.7 and 1.4.8,

P σ [H -1 = H + B + ∪{-1} ] = n(σ) + o(1)
, where

n(σ) = 1/2 if σ ∈ R c , 1/3 if σ ∈ R i . Since λ(σ) = 2 + o(1) if σ ∈ R c , 3 + o(1) if σ ∈ R i ,
and since µ β (σ) = µ β (σ * ), we conclude that

P -1 [H σ = H B + ] = µ β (σ * ) cap(-1, B + ) 1{σ ∈ R a } + o(1)
.

Summing over σ ∈ B + , we conclude that µ β (σ * )/cap(-1, B + ) = |R a | -1 (1 + o(1)),
which completes the proof of the assertion.

It follows from the proof of the previous lemma that for any configuration

σ * ∈ R a , lim β→∞ cap(-1, B + ) µ β (σ * ) = |R a | . (1.26)
Denote by R l , R s the configurations of R a in which the extra particle is attached to one of the longest, shortest sides, respectively, and let

R lc = R l ∩R c , R li = R l ∩R i .
The next lemma is an immediate consequence of Assertions 1.4.7 and 1.4.8. Lemma 1.4.10. For σ ∈ R lc , σ ∈ R li , and σ ∈ R s ,

lim β→∞ P σ [H -1 = H M ] = 1/2 and lim β→∞ P σ [H 0 = H M ] = 1/2 , lim β→∞ P σ [H -1 = H M ] = 1/3 and lim β→∞ P σ [H 0 = H M ] = 2/3 , lim β→∞ P σ [H -1 = H M ] = 1 .
It follows from this assertion that for every σ ∈ R a ,

lim β→∞ P σ [H {-1,0} < H +1 ] = 1 , (1.27) Proof.[Proof of Proposition 1.2.3]
We prove below the first idendity of the proposition. We first claim that cap(-1, {0, +1}) = cap(-1, B + )

σ∈B + P -1 [H σ = H B + ] P σ [H {0,+1} < H -1 ] .
(1.28) Indeed, since starting from -1 the process hits B + before {0, +1}, by the strong Markov property we have that

P -1 [H {0,+1} < H + -1 ] = σ∈B + P -1 [H σ = H + B + ∪{-1} ] P σ [H {0,+1} < H -1 ] .
By (1.10), we may rewrite the previous expression as

P -1 [H B + < H + -1 ] σ∈B + P -1 [H σ = H B + ] P σ [H {0,+1} < H -1 ] .
This proves (1.28) in view of the definition (1.5) of the capacity. By (1.28) and (1.26), for any configuration

σ * ∈ R a , lim β→∞ cap(-1, {0, +1}) µ β (σ * ) = |R a | lim β→∞ σ∈B + P -1 [H σ = H B + ] P σ [H {0,+1} < H -1 ] .
By Lemma 1.4.9, the right hand side is equal to

lim β→∞ σ∈R a P σ [H {0,+1} < H -1 ] . ( 1
cap(-1, 0) µ β (σ * ) = lim β→∞ σ∈R a P σ [H 0 < H -1 ] .
By (1.27), this expression is equal to (1.29). This completes the proof of the assertion.

Proof.[Proof of Proposition 1.2.1] Let q(σ) = P -1 [H σ = H B + ], σ ∈ B + . By Assertion 1.4.6, q(σ) → 0 if σ ∈ R a . Hence, by (1.20), P -1 [H +1 < H 0 ] = σ∈B + q(σ) P σ [H +1 < H 0 ] = σ∈R a q(σ) P σ [H +1 < H 0 ] + o(1)
.

By (1.27), for all σ ∈ R a , P σ [H +1 < H 0 ] = P σ [H +1 < H 0 , H {-1,0} < H +1 ] = P σ [H -1 < H +1 < H 0 ] .
Therefore, by the strong Markov property,

P -1 [H +1 < H 0 ] = P -1 [H +1 < H 0 ] σ∈R a q(σ) P σ [H -1 < H {0,+1} ] + o(1)
.

By Lemma 1.4.10, for σ ∈ R l , lim sup β→∞ P σ [H -1 < H {0,+1}
] ≤ 1/2, which completes the proof of the proposition.

Proof.[Proof of Proposition 1.2.2] We prove the proposition when the chain starts from -1, the argument being analogous when it starts from 0. Since the chains hits B + before reaching 0 and R l , by the strong Markov property,

P -1 [H R l < H 0 ] = σ∈B + P -1 [H σ = H B + ] P σ [H R l < H 0 ] .
By Lemma 1.4.9, this expression is equal to

1 + o(1) 1 |R a | |R l | + σ∈R s P σ [H R l < H 0 ] .
By Assertions 1.4.7 and 1.4.8, for all σ ∈ R s , σ ∈ R

lim β→∞ P σ [H R < H R l ∪{-1,0} ] = 1 , lim β→∞ P σ [H -1 < H R l ∪{0} ] = 1 .
Therefore, for all σ ∈ R s ,

lim β→∞ P σ [H -1 < H R l ∪{0} ] = 1 .
Hence, by the strong Markov property and by the first two identities of this proof,

P -1 [H R l < H 0 ] = 1 + o(1) 1 |R a | |R l | + σ∈R s P -1 [H R l < H 0 ] ,
which completes the proof of the proposition.

Proofs of Propositions 1.2.3 and 1.2.4

We examine in this section the metastable behavior of the Blume-Capel model starting from 0. The main observation is that the energy barrier from 0 to -1 is larger that the one from 0 to +1. We may therefore ignore -1 and argue by symmetry that the passage from 0 to +1 is identical to the one from -1 to 0.

In analogy to the notation introduced right before Assertion 1.4.4, let B 0 be the set of configurations with n 0 (n 0 + 1) sites with spins different from 0, and let R 0 be the subset of B 0 given by

R 0 = σ ∈ {0, +1} Z 2 : {x : σ(x) = 0} forms a n 0 × (n 0 + 1) rectangle .
Denote by R + 0 the set of configurations in {-1, 0, +1} Z 2 in which there are n 0 (n 0 + 1) + 1 spins which are not equal to 0, and in which n 0 (n 0 + 1) spins of magnetization +1 form a n 0 × (n 0 + 1)-rectangle. Finally, let R a 0 ⊂ R + 0 be the set of configurations for which the remaining spin is a +1 spin attached to one of the sides of the rectangle, and let

B + 0 = (B 0 \ R 0 ) ∪ R + 0 . Assertion 1.5.1. For every σ ∈ B + 0 , lim β→∞ P 0 [H σ = H B + 0 ] = |R a 0 | -1 1{σ ∈ R a 0 }. Moreover, if σ * represents a configuration in R a 0 , lim β→∞ cap(0, B + 0 ) µ β (σ * ) = |R a 0 | .
Proof. As in Assertion 1.4.6, we may exclude all configurations σ ∈ {0, +1} Λ L which do not belong to R a 0 . We may also exclude all configurations in B + 0 which have a negative spin since by turning all negative spins into positive spins we obtain a new configurations whose energy is strictly smaller than the one of the original configuration. For the configurations in R a 0 we may apply the arguments presented in the proof of Lemma 1.4.9.

Denote by R c 0 , R i 0 the configurations of R a 0 in which the extra particle is attached to the corner, interior of the rectangle, respectively. Denote by R l 0 , R s 0 the configurations of R a 0 in which the extra particle is attached to one of the longest, shortest sides, respectively, and let

R lc 0 = R l 0 ∩ R c 0 , R li 0 = R l 0 ∩ R i 0 .
The proof of the next assertion is analogous to the one of Lemma 1.4.10 since it concerns configurations with only 0 and +1 spins.

Assertion 1.5.2. For σ ∈ R lc 0 , σ ∈ R li 0 , and σ ∈ R s 0 , lim β→∞ P σ [H 0 = H M ] = 1/2 and lim β→∞ P σ [H +1 = H M ] = 1/2 , lim β→∞ P σ [H 0 = H M ] = 1/3 and lim β→∞ P σ [H +1 = H M ] = 2/3 , lim β→∞ P σ [H 0 = H M ] = 1 .
The next claim follows from the previous two assertions.

Assertion 1.5.3. We have that

lim β→∞ P 0 [H -1 < H +1 ] = 0 .
Proof. Since, starting from 0, the set B + 0 is reached before the process hits {-1, +1}, by the strong Markov property,

lim β→∞ P 0 [H -1 < H +1 ] = lim β→∞ σ∈B + 0 P 0 [H σ = H B + 0 ] P σ [H -1 < H +1 ] .
By Assertion 1.5.1 and by the strong Markov property at time H M , this expression is equal to

lim β→∞ 1 |R a 0 | σ∈R a 0 E σ P σ(H M ) [H -1 < H +1 ] = c 0 lim β→∞ P 0 [H -1 < H +1 ] .
where we applied Assertion 1.5.2 to derive the last identity. In this equation,

c 0 = {|R lc 0 |/2|R a 0 |} + {|R li 0 |/3|R a 0 |} < 1.
This completes the proof of the assertion. Proof.[Proof of Proposition 1.2.3, Part B] The proof is similar to the one of Part A, presented in the previous section. As in (1.28), we have that cap(0, {-1, +1}) = cap(0, B + 0 )

σ∈B + 0 P 0 [H σ = H B + 0 ] P σ [H {-1,+1} < H 0 ] .
Hence, by Assertion 1.5.1, for any configuration Proof. We repeat the proof of the part A of Proposition 1.2.3 up (1.29) to obtain that

σ * ∈ R a 0 , lim β→∞ cap(0, {-1, +1}) µ β (σ * ) = |R a 0 | lim β→∞ σ∈B + 0 P 0 [H σ = H B + 0 ] P σ [H {-1,+1} < H 0 ] .

By

lim β→∞ cap(-1, +1) µ β (σ * ) = lim β→∞ σ∈R a P σ [H +1 < H -1 ] ,
if σ * represents a configuration in R a . By (1.27), this expression is equal to

lim β→∞ σ∈R a P σ [H 0 < H +1 < H -1 ] = lim β→∞ P 0 [H +1 < H -1 ] σ∈R a P σ [H 0 < H {-1,+1} ] .
where we used the strong Markov property in the last step. By Lemma 1.4.10 and Assertion 1.5. To complete the proof, it remains to recall (1.30).

We turn to the proof of Proposition 1.2.4. We first show that the assumptions of Lemma 1.

∈ M. If H(σ) ≤ H(η), then H(ξ σ,M ) -H(σ) < H(ξ σ,η ) -H(η).
Proof. We claim that for any configuration σ ∈ M, H(ξ σ,M ) -H(σ) ≤ 2 -h. To prove this claim it is enough to exhibit a self-avoiding path from σ to M whose energy is kept below H(σ) + 2 -h. This is easy. Starting from σ we may first reach the set S of stable configurations through a path whose energy does not increase. Denote by σ the configuration in S attained through this path. From σ we may reach the set M by removing all small droplets (the ones whose smaller side has length n 0 or less) and by increasing the large droplets (the ones whose both sides have length at least n 0 + 1) in such a way that the energy remains less than or equal to H(σ ) + 2 -h. This proves the claim.

On the other hand, since H(ζ) ≥ H(η) + 4 -h for any configuration ζ which differs from η at one site, H(ξ σ,η ) -H(η) ≥ 4 -h, which proves the assertion. Assertion 1.5.7. We have that

lim β→∞ M (-1) P -1 [H 0 < H + {-1,+1} ] cap(-1, {0, +1}) = 1 . Proof. Fix σ * in R a . In view of Proposition 1.2.3, it is enough to show that lim β→∞ M (-1) P -1 [H 0 < H + {-1,+1} ] µ β (σ * ) = 4(2n 0 + 1) 3 |Λ L | .
In the proof of Proposition

E 0 [H +1 ] = 1 + o(1) 1 cap(0, +1) µ β (0) + µ β (-1)P -1 [H 0 < H +1 ] .
Since the second term in the expression inside braces is bounded by µ β (-1) ≺ µ β (0), the expectation is equal to (1 + o( 1))µ β (0)/cap(0, +1). By (1.30), we may replace cap(0, +1) by cap(0, {-1, +1}). On the other hand, by Proposition 1.2.3,

lim β→∞ µ β (0) cap(0, {-1, +1}) cap(-1, {0, +1}) µ β (-1) = 1 .
This completes the proof of the proposition in the case in which the chain starts from 0.

We turn to the case in which the chain starts from -1. By Lemma 1.3.3,

E -1 [H +1 ] = 1 + o(1) 1 cap(-1, +1) µ β (-1) + µ β (0)P 0 [H -1 < H +1 ] .
By (1.10), by reversibility and by definition of the capacity,

µ β (0) P 0 [H -1 < H +1 ] = M (0) P 0 [H -1 < H + {0,+1} ] λ β (0) P 0 [H {-1,+1} < H + 0 ] = M (-1) P -1 [H 0 < H + {-1,+1} ] λ β (0) P 0 [H {-1,+1} < H + 0 ] = µ β (0) M (-1) P -1 [H 0 < H + {-1,+1} ] cap(0, {-1, +1}) •
Hence, by Assertion 1.5.7,

E -1 [H +1 ] = 1 + o(1) µ β (-1) cap(-1, +1) 1 + µ β (0) cap(-1, {0, +1}) µ β (-1) cap(0, {-1, +1}) .
To complete the proof it remains to recall the statements of Proposition 1.2.3 and Assertion 1.5.4.

The hitting time of 0 starting from -1

We prove in this section Proposition 1.2.5. By Lemma 1.3.3 and Assertion 1.5.6,

E -1 [H 0 ] = 1 + o(1) 1 cap(-1, 0) µ β (-1) + µ β (+1)P +1 [H -1 < H 0 ] .
By (1.10) and the first identity in (1.11), and by reversibility, the second term inside braces is equal to

µ β (+1) M (+1) P +1 [H -1 < H + {0,+1} ] cap(+1, {-1, 0}) = µ β (+1) M (-1) P -1 [H +1 < H + {-1,0} ] cap(+1, {-1, 0})
•

By the first identity in (1.11), this expression is equal to

µ β (+1) cap(-1, {0, +1}) cap(+1, {0, -1}) P -1 [H +1 < H 0 ] .
By Assertion 1.5.5 and Proposition 1.2.3, we may replace the ratio of the capacities by µ β (-1)/µ β (0). Hence,

E -1 [H 0 ] = 1 + o(1) µ β (-1) cap(-1, 0) 1 + µ β (+1) µ β (0) P -1 [H +1 < H 0 ] .
By Lemma 1.6.1 below, this expression is equal to

E -1 [H 0 ] = 1 + o(1) µ β (-1) cap(-1, 0) µ β (+1) µ β (0) P -1 [H +1 < H 0 ] .
By Assertion 1.4.11, we may replace in the right hand side of the previous formula cap(-1, 0) by cap(-1, {0, +1}). This proves the first assertion of the proposition in view of the definition of θ β . The second assertion of the proposition is the content of Lemma 1.6.1.

Lemma 1.6.1. We have that

lim β→∞ µ β (+1) µ β (0) P -1 [H +1 < H 0 ] = ∞ .
The proof of this lemma is divided in several assertions. By (1.11), and by the definition of the capacity,

P -1 [H +1 < H 0 ] = µ β (-1) λ β (-1) P -1 [H +1 < H + {-1,0} ] cap(-1, {0, +1}) • (1.31)
We estimate the probability appearing in the numerator. This is done by proposing a path from -1 to +1 which does not visit 0. The obvious path is the optimal one from -1 to 0 juxtaposed with the optimal one from 0 +1, modified not to visit 0.

We describe the path in S, the set of stable configurations introduced in (1.25). Let ξ 0 ∈ S be the configuration formed by a L × (L -2) band of 0-spins and a L × 2 band of -1 spins. The first piece of the path, denoted by γ 0 , connects -1 to ξ 0 . It is formed by creating and increasing a droplet of 0-spins in a sea of -1-spins.

Let γ 0 = (-1 = η 0 , . . . , η N = ξ 0 ), where -N = 2(L -3), η 1 is a 2 × 2 square of 0-spins in a background of negative spins, -For k < N -1, η k+1 is obtained from η k adding a line of 0-spins to transform a j × j-square of 0-spins into a (j + 1) × j-square of 0-spins, or to transform (j + 1) × j-square of 0-spins into a j × j-square of 0-spins.

Note that η N is obtained from η N -1 transforming a (L -2) × (L -2) rectangle into a L × (L -2) band.
Let ξ 1 ∈ S be the configuration formed by a 2 × 2 square of +1-spins in a background of 0-spins. The last piece of the path, denoted by γ 1 , connects ξ 0 to +1 and is constructed in a similar way as γ 0 so that γ 1 = (ξ 1 = ζ 0 , . . . , ζ N = +1). Note that the length of γ 1 is the same as the one of γ 0 .

Denote by q(η, ξ) the jump probabilities of the trace of σ(t) on S : q(η, ξ

) = P η [H ξ = H S\{η} ]. Let q(γ 0 ) = N -1 k=0 q(η k , η k+1 ) , q(γ 1 ) = N -1 k=0 q(ζ k , ζ k+1 ) , so that P -1 H +1 < H + {-1,0}
≥ q(γ 0 ) q(ξ 0 , ξ 1 ) q(γ 1 ) .

(1.32)

We estimate the three terms on the right hand side.

Assertion 1.6.2. There exists a positive constant c 0 , independent of β, such that q(γ 0 ) ≥ c 0 e -β{4(n 0 -1)-[n 0 (n 0 +1)-2]h} .

Proof. By the arguments presented in the proof of Assertion 1.4.8, there exists a positive constant c 0 , independent of β, such that q(η k , η k+1 ) ≥ c 0 if k ≥ 2n 0 -1. Thus,

q(γ 0 ) ≥ c 0 2(n 0 -1) k=0 q(η k , η k+1 ) ,
and η 2n 0 -1 is a (n 0 + 1) × (n 0 + 1) square of 0 spins in a sea of -1-spins.

Denote by λ S the holding rates of the trace of σ(t) on S, by µ S the invariant probability measure, and let M S (η) = λ S (η)µ S (η). The measure M S is reversible for the dicrete-time chain which jumps from η to ξ with probability q(η, ξ).

By the proof of Assertion 1.4.8, there exists a positive constant c 0 , independent of β, such that q(η k+1 , η k ) ≥ c 0 if k < 2(n 0 -1). Thus, multiplying and dividing by M S (-1), by reversibility

2(n 0 -1) k=0 q(η k , η k+1 ) = M S (η 2(n 0 -1) ) M S (-1) 2n 0 -3 k=0 q(η k+1 , η k ) q(η 2(n 0 -1) , η 2n 0 -1 ) ≥ c 0 M S (η 2(n 0 -1) ) M S (-1) q(η 2(n 0 -1) , η 2n 0 -1 ) ,
where the configuration η 2(n 0 -1) is a (n 0 + 1) × n 0 rectangle of 0-spins.

Recall that M S (η) = µ S (η) λ S (η). Since µ S (η) = µ β (η)/µ β (S), by [2, Proposition 6.1], for any η ∈ S,

M S (η) = µ β (η) µ β (S) λ β (η) P η H S\{η} < H + η = cap(η, S \ {η}) µ β (S) • (1.33)
We claim that

M S (η 2(n 0 -1) ) q(η 2(n 0 -1) , η 2n 0 -1 ) ≥ c 0 µ S (η 2(n 0 -1) ) e -β(2-h) . (1.34)
To keep notation simple, let η = η 2(n 0 -1) , ξ = η 2n 0 -1 . By definition of q and by the first equality in (1.11), the jump probability appearing on the left hand side is equal to

P η H ξ = H S\{η} = µ β (η) λ β (η) P η H ξ = H + S cap(η, S \ {η}) •
The denominator cancels the numerator in (1.33). On the other hand, to reach ξ from η without returning to η, the simplest way consists in creating a 0-spin attached to the longer side of the rectangle and to build a line of 0-spins from this first one. Only the first creation has a cost which vanishes as β ↑ ∞. Hence,

λ β (η) P η H ξ = H + S ≥ c 0 R β (η, η )
where η is a critical configuration in R l . This completes the proof of (1.34) since R β (η, η ) = exp{-β(2 -h)}.

It remains to estimate M S (-1). Recall (1.23). Since ξ -1,S\{-1} is the configuration with three 0-spins included in a 2 × 2 square, cap(-1, S \ {-1}) ≤ C 0 exp{-β[8 -3h]}µ β (-1). Hence, by (1.33), M S (-1) ≤ C 0 e -β(8-3h) µ S (-1) .

(1.35)

Putting together all previous estimates, we obtain that (8-3h) , which completes the proof of the assertion in view of the definition of η 2(n 0 -1) .

q(γ 0 ) ≥ c 0 µ β (η 2(n 0 -1) ) µ β (-1) e -β(2-h) e β
Next result is proved similarly.

Assertion 1.6.3. There exists a positive constant c 0 , independent of β, such that q(γ 1 ) ≥ c 0 e -β{4(n 0 -1)-[n 0 (n 0 +1)-2]h} .

We turn to the probability q(ξ 0 , ξ 1 ). Recall that ξ 0 is the configuration formed by a L × (L -2) band of 0-spins and a L × 2 band of -1 spins, and that ξ 1 is the configuration formed by a 2 × 2 square of +1-spins in a background of 0-spins. Assertion 1.6.4. There exists a positive constant c 0 , independent of β, such that

q(ξ 0 , ξ 1 ) ≥ c 0 e -2β[2-h] .
Proof. By definition of q and by (1.10),

q(ξ 0 , ξ 1 ) = P ξ 0 H ξ 1 = H S\{ξ 0 } = µ β (ξ 0 ) λ β (ξ 0 ) P ξ 0 H ξ 1 = H + S cap(ξ 0 , S \ {ξ 0 }) • We claim that P ξ 0 H ξ 1 = H + S ≥ c 0 e -2β[2-h] . (1.36)
To estimate this probability, we propose a path γ 3 from ξ 0 to ξ 1 which avoids S. The path consists in filling the -1-spins with 0-spins, until one -1-spin is left. At this point, to avoid the configuration 0, we switch this -1-spin to +1. To complete the path we create a 2 × 2 square of +1-spins from the first +1-spin, as in the optimal path from 0 to ξ 1 . Hence, γ 3 as length 2L+3. Denote this path by γ 3 = (ξ 0 = η 0 , η 1 , . . . , η 2L+3 = ξ 1 ). From η 0 to η 2L-2 the next configurations is obtained by flipping a -1 spin to a 0spin as in an optimal path from ξ 0 to 0. In this piece of the path, all jumps have a probability bounded below by a positive constant. Therefore, there exists a positive constant c 0 , independent of β, such that

P ξ 0 H ξ 1 = H + S ≥ 2L+2 j=0 p β (η j , η j+1 ) ≥ c 0 2L+2 j=2L-1 p β (η j , η j+1 ) .
The first and the last probabilities in this product, p β (η 2L-1 , η 2L ) and p β (η 2L+2 , η 2L+3 ), are also bounded below by a positive constant. The other ones can be estimated easily, proving (1.36).

By (1.23), cap(ξ 0 , S \ {ξ 0 })/µ β (ξ 0 ) is bounded above by C 0 exp{-β[2 -h]}, while an elementary computation shows that λ β (ξ 0 ) is bounded below by c 0 exp{-β[2-h]}. This completes the proof of the assertion. By (1.32) and Assertions 1.6.2, 1.6.3 and 1.6.4,

P -1 H +1 < H + {-1,0}
≥ c 0 e -2β{2(2n 0 -1)-[n 0 (n 0 +1)-1]h} .

(1.37)

Proof.[Proof of Lemma 1.6.1] Since λ(-1) ≥ c 0 e -β [4-h] , by (1.31), (1.37),

P -1 [H +1 < H 0 ] ≥ c 0 µ β (-1) cap(-1, {-1, 0}) e -β{8n 0 -[2n 0 (n 0 +1)-1]h} .
Therefore, by Proposition 1.2.3,

µ β (+1) µ β (0) P -1 [H +1 < H 0 ] ≥ c 0 e βL 2 e -β{4(n 0 -1)-[n 0 (n 0 +1)-2]h} .
It remains to show that L 2 > 4(n 0 -1) -[n 0 (n 0 + 1) -2]h. By definition of n 0 , n 0 h > 2 -h, so that 4(n 0 -1) -[n 0 (n 0 + 1) -2]h ≤ 2n 0 -6 + hn 0 + 3h. As hn 0 < 2 and h < 1, this expression is less than or equal to 2n 0 . This expression is smaller than L 2 because L ≥ 2 and L > n 0 .

Proof of Theorem 1.2.6

The statement of Theorem 1.2.6 follows from Proposition 1.7.3 and Lemma 1.7.6 below and from Theorem 5.1 in [START_REF] Landim | A topology for limits of Markov chains[END_REF]. We start deriving some consequences of the assumption (1.8). Clearly, it follows from (1.8) and from (1.23) that for all η ∈ M and σ

∈ V η , σ = η, µ β (η) cap(σ, η) ≺ θ β . (1.38) Assertion 1.7.1. For all η ∈ M and σ ∈ V η , σ = η, µ β (σ) ≺ µ β (η) , cap(η, M \ {η}) ≺ cap(σ, η) .
Proof. The first bound is a straightforward consequence of the hypothesis H(σ) > H(η). In view of (1.23), to prove the second bound we have to show that H(σ, η) < H(η, M \ {η}). The case η = -1 is a consequence of the second hypothesis in (1.8), as the case η = 0 if one recalls (1.9). It remains to consider the case η = +1. By condition (1.8) and (1.9),

H(σ, +1) < H(+1) + H(0, {-1, +1}) -H(0) < H(0, {-1, +1}) .
By Assertion 1.5.5 and by (1.23), H(0, {-1, +1}) = H(+1, {-1, 0}). Therefore,

H(σ, +1) < H(+1, {-1, 0}) ,
which proves the second claim of the assertion.

Assertion 1.7.2. For all η = ξ ∈ M and for all σ ∈ V η , σ ∈ V ξ , cap(σ, σ ) ≈ cap(η, ξ) .

Proof. Fix η = ξ ∈ M and σ ∈ V η , σ ∈ V ξ . We need to prove that H(η, ξ) = H(σ, σ ). On the one hand, by definition, H(σ, σ ) ≤ max{H(σ, η), H(η, η ), H(η , σ )}. By the proof of Assertion 1.7.1, H(σ, η) < H(η, M \ {η}), with a similar inequality replacing σ, η by σ , η , respectively. Since H(A, B) is decreasing in each variable, H(η, M\{η}) and H(η , M \ {η }) are less than or equal to H(η, η ), which shows that H(σ, σ ) ≤ H(η, η ).

Conversely, H(η, η ) ≤ max{H(η, σ), H(σ, σ ), H(σ , η )}. By the previous paragraph, H(η, σ) < H(η, η ) and H(σ , η ) < H(η, η ) so that H(η, η ) ≤ H(σ, σ ). This completes the proof of the assertion.

We conclude this preamble with two simple remarks. Fix η ∈ M and σ ∈ V η . By (1.13) and Assertion 1.7.2,

cap(σ, ∪ ξ =η V ξ ) ≈ max σ ∈∪ ξ =η V ξ cap(σ, σ ) ≈ max ξ∈M\{η} cap(η, ξ) .
Applying (1.13) once more, we conclude that cap(σ, ∪ ξ =η V ξ ) ≈ cap(η, M \ {η}) .

(1.39)

In particular, by Assertion 1.7.1,

lim β→∞ cap(σ, ∪ ξ =η V ξ ) cap(σ, η) = 0 . (1.40)
Denote by σ A (t), A ⊂ Ω, the trace of σ(t) on A. By [2, Proposition 6.1], σ A (t) is a continuous-time Markov chain. Moreover, for B ⊂ A, σ B (t) is the trace of σ A (t) on B. When A = M, we represent σ A (t) by η(t). Denote R A β (σ, σ ), σ = σ ∈ A, the jump rates of the Markov chain σ A (t).

Recall the definition of the map π : M → {-1, 0, 1}, introduced just before the statement of Theorem 1.2.6. Denote by ψ = ψ V : V → {-1, 0, 1} the projections defined by

ψ(σ) = π(η) if σ ∈ V η : ψ(σ) = η∈M π(η) 1{σ ∈ V η } .
Recall also the definition of the time-scale θ β introduced in (1.6).

Proposition 1.7.3. As β ↑ ∞, the speeded-up, hidden Markov chain ψ(σ V (θ β t)) converges to the continuous-time Markov chain X(t) introduced in Theorem 1.2.6.

We first prove Proposition 1.7.3 in the case where the wells V η are singletons : V η = {η}. In this case, ψ is a bijection, and ψ(η(t)) is a Markov chain on {-1, 0, 1}.

r β (η, ξ) = θ β µ β (η) cap(η, M \ {η}) P η [H ξ < H M\{η,ξ} ] .
For η = +1 and ξ = -1, 0, by Assertion 1.5.5 and by Proposition 1.2.3,

lim β→∞ r β (+1, ξ) ≤ lim β→∞ θ β µ β (+1) cap(+1, M \ {+1}) = lim β→∞ µ β (0) µ β (+1) = 0 .
On the other hand, by Proposition 1.2.3,

lim β→∞ θ β µ β (0) cap(0, M \ {0}) = 1 , while θ β cap(-1, M \ {-1})/µ β (-1) = 1. Furthermore,
P -1 [H +1 < H 0 ] = lim β→∞ P 0 [H -1 < H +1 ] = 0 .
This yields (1.41) and completes the proof of the lemma.

Denote by P V σ , σ ∈ V, the probability measure on the path space D(R + , V) induced by the Markov chain σ V (t) starting from σ. Expectation with respect to P V σ is represented by E V σ . Clearly, for any disjoint subsets A, B of V,

P V σ [H A < H B ] = P σ [H A < H B ] . (1.42)
The hitting time of a subset A of V by the trace chain σ V can be represented in terms of the original chain σ(t). Under P σ ,

H V A = inf{t > 0 : σ V (t) ∈ A} = H A 0 1{σ(t) ∈ V} dt . (1.43) Let V(η) = Vη = ζ =η V ζ .
Denote by {T j : j ≥ 0} the jump times of the hidden chain ψ(σ V (t)) :

T 0 = 0 , T j+1 = inf{t ≥ T j : σ V (t) ∈ V(σ V (T j )) , j ≥ 0 ,
Similarly, denote by {τ j : j ≥ 0} the successive jump times of the chain η(t).

Lemma 1.7.5. Fix σ ∈ V -1 . There exists a sequence ε β → 0 such that for j = 1, 2

lim β→∞ P V σ |τ j -T j | ≥ θ β ε β = 0 , lim β→∞ P V σ T 3 -T 2 ≤ θ β ε -1 β = 0 . (1.44)
Moreover,

lim β→∞ P V σ σ(T 1 ) ∈ V 0 = 0 , lim β→∞ P V σ σ(T 2 ) ∈ V +1 = 0 . (1.45)
Proof. Fix a configuration σ ∈ V -1 . By (1.42), (1.12) and (1.40), lim sup

β→∞ P V σ H V(-1) < H -1 ≤ lim β→∞ cap(σ, V(-1)) cap(σ, -1) = 0 . (1.46)
On the other hand, under P V σ ,

τ 1 = H 0,+1 H -1 1{σ(s) = -1} ds .
Hence, under P V σ and on the event {H -1 < H V(-1) } we have that

T 1 = H -1 + H( V-1 ) H -1 1{σ(s) = -1} + 1{σ(s) = -1} ds = τ 1 + H -1 + H( V-1 ) H -1 1{σ(s) = -1} ds - H 0,+1 H( V-1 )
1{σ(s) = -1} ds .

It remains to estimate the last three terms.

To bound the first term, by (1.42), (1.12), and (1.40), lim sup

β→∞ P V σ [H V-1 < H -1 ] = lim sup β→∞ P σ [H V-1 < H -1 ] ≤ lim β→∞ cap(σ, V-1 ) cap(σ, -1) = 0 .
Hence, to prove that

P V σ [H -1 > θ β ε β ] → 0, it is enough to show that lim β→∞ P V σ [H V-1 ∪{-1} > θ β ε β ] = 0 .
By Tchebycheff inequality, by Lemma 6.9 and Proposition 6.10 in [START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains[END_REF], and by (1.42), the previous probability is less than or equal to

1 θ β ε β 1 cap(σ, V-1 ∪ {-1}) η∈V µ β (η) P η [H σ < H V-1 ∪{-1} ] .
By definition of θ β , since the capacity is monotone, and since we may restrict the sum to V -1 , the previous expression is less than or equal to

1 ε β cap(-1, {0, +1}) cap(σ, -1) µ β (V -1 ) µ β (-1) • By Assertion 1.7.1, µ β (V -1 )/µ β (-1
) is bounded and cap(-1, {0, +1})/cap(σ, -1) vanishes as β ↑ ∞. Hence, the previous expression converges to 0 for every sequence ε β which does not decrease too fast.

We turn to the second term of the decomposition of T 1 . By the strong Markov property and by (1.43), we need to estimate,

P V -1 H( V-1 ) 0 1{σ(s) = -1} ds > θ β ε β = P -1 H( V-1 ) 0 1{σ(s) ∈ V \ {-1}} ds > θ β ε β .
By Tchebycheff inequality and by [2, Proposition 6.10], the previous probability is less than or equal to

1 θ β ε β 1 cap(-1, V-1 ) η∈V\{-1} µ β (η) P η [H -1 < H V-1 ] .
By (1.39), cap(-1, V-1 ) ≈ cap(-1, {0, +1}). Hence, by definition of θ β , and since the sum can be restricted to the set V -1 \ {-1}, the previous expression is less than or equal to

C 0 ε β 1 µ β (-1) µ β (V -1 \ {-1})
for some finite constant C 0 . By Assertion 1.7.1, the ratio of the measures vanishes as β ↑ ∞. In particular, the previous expression converges to 0 as β ↑ ∞ if ε β does not decrease too fast. The third term in the decomposition of T 1 is absolutely bounded by H 0,+1 -H( Vη ) and can be handled as the first one. This proves the first assertion of (1.44) for j = 1.

In a similar way one proves that T 2 -T 1 is close to τ 2 -τ 1 . The first assertion of (1.44) for j = 2 follows from this result and from the bound for T 1 -τ 1 . The details are left to the reader.

We turn to the proof of the first assertion in (1.45). Since V-1 = V 0 ∪ V +1 ,

P V σ σ(T 1 ) ∈ V 0 = P σ σ(H) ∈ V +1 ,
where H = H( V-1 ). We may rewrite the previous probability as

P σ σ(H) ∈ V +1 , H 0 < H +1 + P σ σ(H) ∈ V +1 , H 0 > H +1 .
Both expression vanishes as β ↑ ∞.

P σ H 0 < H +1 .
This expression converges to 0 as β ↑ ∞ because V +1 is contained in the basin of attraction of +1. The proof of the second assertion in (1.45) is similar and left to the reader. We finally examine the second assertion of (1.44). By the second assertion of (1.45), it is enough to prove that

lim β→∞ P V σ T 3 -T 2 ≤ θ β ε -1 β , σ(T 2 ) ∈ V +1 = 0 .
By the strong Markov property, this limit holds if

lim β→∞ max σ ∈V +1 P V σ T 1 ≤ θ β ε -1 β = 0 .
Since V +1 is contained in the basin of attraction of +1, it is enough to show that

lim β→∞ max σ ∈V +1 P V σ T 1 ≤ θ β ε -1 β , H +1 < T 1 = 0 .
On the event

{H +1 < T 1 }, {T 1 ≤ θ β ε -1 β } ⊂ {T 1 • θ H +1 ≤ θ β ε -1 β }
, where {θ t : t ≥ 0} represents the semigroup of time translations. In particular, by the strong Markov property, we just need to show that

lim β→∞ P V +1 T 1 ≤ θ β ε -1 β = 0 .
Let {e j : j ≥ 1} be the length of the sojourn times at +1. Hence, {e j : j ≥ 1} is a sequence of i.i.d. exponential random variables with parameter λ(+1). Denote by A the set of configurations with at least n 0 (n 0 + 1) sites with spins not equal to +1. Each time the process leaves the state +1 it attempts to reach A before it returns to +1. Let δ be the probability of success :

δ = P V +1 H A < H + +1 .
Let N ≥ 1 be the number of attempts up to the first success so that 1≤j≤N e j represents the total time the process σ(t) remained at +1 before it reached A. It is clear that under

P V +1 , N j=1 e j ≤ T 1 ,
and that N is a geometric random variable of parameter δ independent of the sequence {e j : j ≥ 1}. In view of the previous inequality, it is enough to prove that

lim β→∞ P +1 N j=1 e j ≤ θ β ε -1 β = 0 .
The previous probability is less than or equal to

λ(+1) θ β ε -1 β P V +1 H A < H + +1 = 1 ε β θ β µ β (+1) cap(A, +1) .
Since θ β cap(A, +1)/µ β (+1) ≺ 1, the previous expression vanishes if ε β does not decrease too fast to 0. This completes the proof of the lemma.

Proof.[Proof of Proposition 1.7.3] The assertion of the proposition is a straightforward consequence of Lemmas 1.7.4 and 1.7.5. Fix σ ∈ V -1 and recall the notation introduced in Lemma 1.7.5.

Let A = {σ(T 1 ) ∈ V 0 } ∩ {σ(T 2 ) ∈ V +1 }. By (1.45), P V σ [A c ] → 0. On the set A, ψ(σ V (θ β t)) = -1{t < T 1 /θ β } + 1{T 2 /θ β ≤ t < T 3 /θ β }
By Lemma 1.7.4, (τ 1 /θ β , (τ 2 -τ 1 )/θ β ) converges to a pair of independent, mean 1, exponential random variables. Hence, by (1.44), (T 1 /θ β , (T 2 -T 1 )/θ β , (T 3 -T 2 )/θ β ) converges in distribution to (e 1 , e 2 , ∞), where (e 1 , e 2 ) is a pair of independent, mean 1, exponential random variables. This completes the proof. Proof. Fix ξ ∈ V +1 . On the one hand, by [2, Proposition 6.10],

1 θ β E ξ H +1 0 1{σ(s) ∈ ∆} ds ≤ 1 θ β µ β (+1) cap(ξ, +1) µ β (∆) µ β (+1) •
This expression vanishes as β ↑ ∞ because, by (1.38), µ β (+1)/cap(ξ, +1) θ β , and because µ β (∆) ≺ µ β (+1), as +1 is the unique ground state.

On the other hand, by the strong Markov property,

1 θ β E ξ tθ β H +1 1{σ(s) ∈ ∆} ds ≤ 1 θ β E +1 tθ β 0 1{σ(s) ∈ ∆} ds .
Therefore, to prove the lemma for ξ ∈ V +1 it is enough to show that

lim β→∞ E +1 t 0 1{σ(sθ β ) ∈ ∆} ds = 0 .
This last assertion follows from Lemma 1.7.7 below. Similar arguments permit to reduce the statement of the lemma for ξ ∈ V 0 (resp. ξ ∈ V -1 ) to the verification that

E +1 t 0 1{σ(sθ β ) ∈ ∆ * } ds ≤ 1 µ β (+1) σ∈Ω µ β (σ) E σ t 0 1{σ(sθ β ) ∈ ∆ * } ds .
Since µ β is the stationary state, the previous expression is equal to

tµ β (∆ * ) µ β (+1) ,
which vanishes as β → ∞ because +1 is the unique ground state. We turn to the case ξ = 0. We first claim that

lim β→∞ E 0 1 θ β H {-1,+1} 0 1{σ(s) ∈ ∆ * } ds = 0 . (1.47)
Indeed, by [2, Proposition 6.10], the previous expectation is equal to

1 θ β V 1{∆ * } µ β cap(0, {-1, +1}) = µ β (0) θ β cap(0, {-1, +1}) V 1{∆ * } µ β µ β (0) , where V is the harmonic function V (σ) = P σ [H 0 < H {-1,+1} ]
. By definition of θ β , the first fraction on the right hand side is bounded. It remains to estimate the second fraction, which is equal to 1 µ β (0)

σ∈∆ * 0 µ β (σ)P σ [H 0 < H {-1,+1} ] ,
where ∆ * 0 = {σ ∈ ∆ * : µ β (σ) µ β (0)}. By (1.12), this sum is less than or equal to

σ∈∆ * 0 cap(σ, 0) µ β (0) µ β (σ) cap(σ, M)
.

Each term of this sum vanishes as β ↑ ∞. Indeed, as σ belongs to ∆ * 0 , to reach σ from 0 the chain has to escape from the well of 0 so that cap(σ, 0)/µ β (0) ≈ θ -1 β . On the other hand, µ β (σ)/cap(σ, M) is the time scale in which the process reaches one of the configurations in M starting from σ, a time scale of smaller order than the one in which it jumps between configurations in M.

By (1.47), to prove the lemma for ξ = 0, we just have to show that

lim β→∞ E 0 1 θ β tθ β 0 1{σ(s) ∈ ∆ * } ds 1{H {-1,+1} ≤ tθ β } = 0 . Since P 0 [H -1 < H +1 ] → 0,
E +1 1 θ β tθ β 0 1{σ(s) ∈ ∆ * } ds .
By the first part of the proof this expectation vanishes as β ↑ ∞.

It remains to consider the case ξ = -1. As in the case ξ = 0, we first estimate the expectation in (1.47), with H {0,+1} instead of H {-1,+1} . Then, we repeat the arguments presented for ξ = 0, with obvious modifications, to reduce the case ξ = -1 to the case ξ = 0, which has already been examined.

We conclude this section proving the assertion of Remark 1.2.7. Fix η ∈ M, σ ∈ V η , σ = η. By (1.12), (1.13) and Assertion 1.7.2,

P σ [H M\{η} < H η ] ≤ cap(σ, M \ {η}) cap(σ, η) ≈ ξ∈M\{η} cap(σ, ξ) cap(σ, η) ≈ ξ∈M\{η} cap(η, ξ) cap(σ, η) •
(1.48) By monotonicity of the capacity, the previous expression is bounded by 2cap(η, M \ {η})/cap(σ, η), which vanishes in view of Assertion 1.7.1. Acknowledgements. The authors wish to thank O. Benois and M. Mourragui for fruitful discussions, and one of the referees for her/his careful reading which permitted to improve the presentation of the article.

Chapitre 2

Metastability of the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field on a large torus 2.1 Introduction

The Blume-Capel model is a two dimensional, nearest-neighbor spin system where the single spin variable takes three possible values : 1, 0 and +1. One can interpret it as a system of particles with spins. The value 0 of the spin at a lattice site corresponds to the absence of particles, whereas the values ±1 correspond to the presence of a particle with the respective spin.

Denote by T L = {1, . . . , L} the discrete, one-dimensional torus of length L, and let Λ L = T L × T L , Ω L = {-1, 0, 1} Λ L . Elements of Ω L are called configurations and are represented by the Greek letter σ. For x ∈ Λ L , σ(x) ∈ {-1, 0, 1} stands for the value at x of the configuration σ and is called the spin at x of σ.

We consider in this article a Blume-Capel model with zero chemical potential and a small positive magnetic field. Fix an external field 0 < h < 2, and denote by H = H L,h : Ω L → R the Hamiltonian given by

H(σ) = (σ(y) -σ(x)) 2 -h x∈Λ L σ(x), (2.1)
where the first sum is carried over all unordered pairs of nearest-neighbor sites of Λ L . We assumed that h < 2 for the configuration whose 0-spins form a rectangle in a background of -1 spins to be a local minima of the Hamiltonian.

The continuous-time Metropolis dynamics at inverse temperature β is the Markov chain on Ω L , denoted by {σ t : t ≥ 0}, whose infinitesimal generator L β acts on

functions f : Ω L → R as (L β f )(σ) = x∈Λ L R β (σ, σ x,+ ) [f (σ x,+ ) -f (σ)] + x∈Λ L R β (σ, σ x,-) [f (σ x,-) -f (σ)] .
In this formula, σ x,± represents the configuration obtained from σ by modifying the spin at x as follows,

σ x,± (z) := σ(x) ± 1 mod 3 if z = x , σ(z) if z = x ,
where the sum is taken modulo 3, and the jump rates R β are given by

R β (σ, σ x,± ) = exp -β H(σ x,± ) -H(σ) + , x ∈ Λ L ,
where a + , a ∈ R, stands for the positive part of a : a + = max{a, 0}. We often write R instead of R β . Denote by µ β the Gibbs measure associated to the Hamiltonian H at inverse temperature β,

µ β (σ) = 1 Z β e -βH(σ) , (2.2)
where Z β is the partition function, the normalization constant which turns µ β into a probability measure. We often write µ instead of µ β .

Clearly, the Gibbs measure µ β satisfies the detailed balance conditions

µ β (σ) R β (σ, σ x,± ) = min µ β (σ) , µ β (σ x,± ) = µ β (σ x,± ) R β (σ x,± , σ) , σ ∈ Ω L , x ∈ Λ L ,
and is therefore reversible for the dynamics. Denote by -1, 0, +1 the configurations of Ω L with all spins equal to -1, 0, +1, respectively. The configurations -1, 0 are local minima of the Hamiltonian, while the configuration +1 is a global minimum. Moreover, H(0) < H(-1).

The existence of several local minima of the energy turns the Blume-Capel model a perfect dynamics to be examined by the theory developed by Beltrán and Landim in [START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains[END_REF][START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains II, the nonreversible case[END_REF] for metastable Markov chains.

Let M = {-1, 0, +1}, and denote by Ψ :

Ω L → {-1, 0, 1, d} the projection defined by Ψ(σ) = η∈M π(η) 1{σ = η} + d 1{σ ∈ M} ,
where d is a point added to the set {-1, 0, 1} and π : M → {-1, 0, 1} is the application which provides the magnetization of the states -1, 0, +1 : π(-1) = -1, π(0) = 0, π(+1) = 1.

A scheme has been developed in [START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains[END_REF][START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains II, the nonreversible case[END_REF][START_REF] Landim | Metastable Markov chains : from the convergence of the trace to the convergence of the finite-dimensional distributions[END_REF] to derive the existence of a time-scale θ β for which the the finite-dimensional distributions of the hidden Markov chain Ψ(σ(tθ β )) converge to the ones of a {-1, 0, 1}-valued, continuous-time Markov chain. Note that the limiting process does not take the value d.

The approach consists in proving first that in the time-scale θ β the trace of the process σ t on M converges in the Skorohod topology to a continuous-time Markov chain. Then, to prove that in this time scale the time spent on Ω L \ M is negligible. Finally, to show that, at any fixed, the probability to be in Ω L \ M is negligible. This is the content of the main result of this article. We are also able to describe the path which drives the process from the highest local minima, -1 to the ground state +1. We not only characterize the critical droplet but we also describe precisely how it grows until it invades the all space. In this process, we show that starting from -1, the model visits 0 on its way to +1.

We consider in this article the situation in which the length of the torus increases with the inverse of the temperature. The case in which L is fixed has been considered by Cirillo and Nardi [START_REF] Emilio | Relaxation height in energy landscapes : an application to multiple metastable states[END_REF], by us [START_REF] Landim | Metastability of the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field[END_REF] and by Cirillo, Nardi and Spitoni [START_REF] Cirillo | Sum of exit times in series of metastable states in probabilistic cellular automata[END_REF].

The method imposes a limitation on the rate at which the space grows, as we need the energy to prevail over the entropy created by the multitude of configurations. In particular, the conditions on the growth impose that the stationary state restricted to the valleys of -1, 0 or +1, defined at the beginning of the next section, is concentrated on these configurations (cf. (2.5)).

The study of the metastability of the Blume-Capel model has been initiated by Cirillo and Olivieri [START_REF] Emilio | Metastability and nucleation for the Blume-Capel model. Different mechanisms of transition[END_REF] and Manzo and Olivieri [START_REF] Manzo | Dynamical Blume-Capel model : competing metastable states at infinite volume[END_REF]. We refer to these papers for the interest of the model and its role in the understanding of metastability.

Notation and Results

Denote by D(R + , Ω L ) the space of right-continuous functions ω : R + → Ω L with left-limits and by P σ = P β,L σ , σ ∈ Ω L , the probability measure on the path space D(R + , Ω L ) induced by the Markov chain (σ t : t ≥ 0) starting from σ. Sometimes, we write σ(t) instead of σ t .

Denote by H A , H + A , A ⊂ Ω L , the hitting time of A and the time of the first return to A respectively :

H A = inf{t > 0 : σ t ∈ A} , H + A = inf{t > T 1 : σ t ∈ A} , (2.3)
where T 1 represents the time of the first jump of the chain σ t .

Critical droplet. We have already observed that +1 is the ground state of the dynamics and that -1 and 0 are local minima of the Hamiltonian. The first main result of this article characterizes the critical droplet in the course from -1 and 0 to +1. Let n 0 = 2/h , where a stands for the integer part of a ∈ R + . Denote by V -1 the valley of -1. This is the set constituted of all configurations which can be attained from -1 by flipping n 0 (n 0 + 1) or less spins from -1. If after n 0 (n 0 + 1) flips we reached a configuration where n 0 (n 0 + 1) 0-spins form a [n 0 × (n 0 + 1)]-rectangle, we may flip one more spin. Hence, all configurations of V -1 differ from -1 in at most n 0 (n 0 + 1) + 1 sites.

The valley V 0 of 0 is defined in a similar way, a n 0 ×(n 0 +1)-rectangle of +1-spins replace the one of 0-spins. Here and below, when we refer to a [n 0 ×(n 0 +1)]-rectangle, n 0 may be its length or its height.

Denote by R l = R l L the set of configurations with n 0 (n 0 + 1) + 1 0-spins forming, in a background of -1-spins, a n 0 × (n 0 + 1) rectangle with an extra 0-spin attached to the longest side of this rectangle. This means that the extra 0-spin is surrounded by three -1-spins and one 0-spins which belongs to the longest side of the rectangle. The set R l 0 = R l 0,L is defined analogously, the -1-spins, 0-spins being replaced by 0-spins, +1-spins, respectively.

We show in the next theorem that, starting from -1, resp. 0, the process visits R l , resp. R l 0 , before hitting {0, +1}, resp. {-1, +1}. An assumption on the growth of the torus is necessary to avoid the entropy of configurations with high energy to prevail over the local minima of the Hamiltonian. We assume that

lim β→∞ |Λ L | e -2β = 0 . (2.4)
We prove in Lemma 2.8.2 that under this condition,

lim β→∞ µ β (V -1 \ {-1}) µ β (-1) = 0 . (2.5)
Theorem 2.2.1. Assume that 0 < h < 1, that 2/h is not an integer and that (2.4) is in force. Then,

lim β→∞ P -1 [H R l < H {0,+1} ] = 1 , lim β→∞ P 0 [H R l 0 < H {-1,+1} ] = 1 .
On the other hand, under the condition that

lim β→∞ |Λ L | 1/2 e -[(n 0 +1)h-2]β + e -hβ and lim β→∞ |Λ L | 2 e -(2-h)β = 0 , (2.6)
it follows from Proposition 2.5.1 that lim β→∞ inf

η∈R l P η [H {0,+1} < H -1 ] > 0 , lim β→∞ inf ξ∈R l 0 P ξ [H {-1,+1} < H 0 ] > 0 .
The route from -1 to +1. The second main result of the article asserts that starting from -1, the processes visits 0 in its way to +1. Actually, in Section 2.5, we describe in detail how the critical droplet growths until it invades the all space.

Theorem 2.2.2. Assume that 0 < h < 1, that 2/h is not an integer and that condition (2.6) is in force. Then,

lim β→∞ P -1 [H +1 < H 0 ] = 0 and lim β→∞ P 0 [H -1 < H +1 ] = 0 .
The strategy of the proof relies on the assumption that while the critical droplet increases, invading the entire space, nothing else relevant happens in other parts of the torus. A new row or column is added to a supercritical droplet at rate e -(2-h)β /|Λ L |, because e -(2-h)β is the rate at which a negative spin is flipped to 0 when it is surrounded by three negative spins and one 0-spin, and L is the time needed for a rate one asymmetric random walk to reach L starting from the origin. We need L 2 because we need an extra 0-spin to complete a row, and then to repeat this procedure L times for the droplet to fill the torus. This rate has to be confronted to the rate at which a 0-spin appears somewhere in the space. The rate at which a negative spin is flipped to 0 when it is surrounded by four negative spins is e -(4-h)β . Since this may happen at |Λ L | different positions, the method of the proof requires at least |Λ L |e -(4-h)β to be much smaller than e -(2-h)β /|Λ L |, that is, |Λ L | 2 e -2β → 0. This almost explains the main hypothesis of the theorem. The extra conditions appear because we need to take care of other details to lengthy to explain here. Metastability. For two disjoint subsets A, B of Ω L , denote by cap(A, B) the capacity between A and B :

cap(A, B) = σ∈A µ β (σ)λ β (σ) P σ [H B < H + A ] ,
where λ β (σ) = σ ∈Ω L R β (σ, σ ) represents the holding times of the Blume-Capel model. Let

θ β = µ β (-1) cap(-1, {0, +1}) • (2.7)
We prove in Proposition 2.7.3 that under the hypotheses of Theorem 2.2.2 for any configuration

η ∈ R l , lim β→∞ cap(-1, {0, +1}) µ β (η) |Λ L | = 4(2n 0 + 1) 3 .
Theorem 2.2.3. Under the hypotheses of Theorem 2.2.2, the finite-dimensional distributions of the speeded-up, hidden Markov chain X β (t) = Ψ σ(θ β t) converge to the ones of the {-1, 0, 1}-valued, continuous-time Markov chain X(t) in which 1 is an absorbing state, and whose jump rates are given by r(-1, 0) = r(0, 1) = 1 , r(-1, 1) = r(0, -1) = 0 .

Note that the limit chain does not take the value d, in contrast with X β (t) since Ψ(σ) = d for all σ ∈ M.

The paper is organized as follows. In the next section, we recall some general results on potential theory of reversible Markov chains and we prove a lemma on asymmetric birth and death chains which is used later in the article. In Section 2.4, we examine the formation of a critical droplet and, in Section 2.5, the growth of a supercritical droplet. Theorems 2.2.1 and 2.2.2 are proved in Section 2.6. In the following two sections, we prove that the trace of σ t on M converges to a three-state Markov chain and that the time spent outside M is negligible. In the final section we prove Theorem 2.2.3.

Metastability of reversible Markov chains

In this section, we present general results on reversible Markov chains used in the next sections. Fix a finite set E. Consider a continuous-time, E-valued, Markov chain {X t : t ≥ 0}. Assume that the chain X t is irreducible and that the unique stationary state π is reversible.

Elements of E are represented by the letters x, y. Let P x , x ∈ E, be the distribution of the Markov chain X t starting from x. Recall from (2.3) the definition of the hitting time and the return time to a set.

Denote by R(x, y), x = y ∈ E, the jump rates of the Markov chain X t , and let λ(x) = y∈E R(x, y) be the holding rates. Denote by p(x, y) the jump probabilities, so that R(x, y) = λ(x) p(x, y). The stationary state of the embedded discrete-time Markov chain is given by M (x) = π(x) λ(x).

Potential theory. Fix two subsets A, B of E such that A ∩ B = ∅. Recall that the capacity between A and B, denoted by cap(A, B), is given by

cap(A, B) = x∈A M (x) P x [H B < H + A ] . ( 2.8)
Denote by L the generator of the Markov chain X t and by D(f ) the Dirichlet form of a function f : E → R :

D(f ) = - x∈E f (x) (Lf )(x) π(x) = 1 2 x,y π(x) R(x, y) [f (y) -f (x)] 2 .
In this later sum, each unordered pair {a, b} ⊂ E, a = b, appears twice. The Dirichlet principle provides a variational formula for the capacity :

cap(A, B) = inf f D(f ) , (2.9)
where the infimum is carried over all functions f : E → [0, 1] such that f = 1 on A and f = 0 on B. Denote by P the set of oriented edges of

E : P = {(x, y) ∈ E × E : R(x, y) > 0}. An anti-symmetric function φ : P → R is called a flow. The divergence of a flow φ at x ∈ E is defined as (div φ)(x) = y:(x,y)∈P

φ(x, y) .

Let F A,B be the set of flows such that

x∈A (div φ)(x) = 1 , y∈B (div φ)(y) = -1 , (div φ)(z) = 0 , z ∈ A ∪ B .
The Thomson principle provides an alternative variational formula for the capacity :

1 cap(A, B) = inf φ∈F A,B 1 2 (x,y)∈P 1 π(x) R(x, y) φ(x, y) 2 . ( 2.10)
We refer to [START_REF] Bovier | of Grundlehren der Mathematischen Wissenschaften[END_REF] for a proof of the Dirichlet and the Thomson principles.

In the Blume-Capel model, by definition of the rate function R β (σ, σ x,± ),

µ β (σ) R β (σ, σ x,± ) = µ β (σ) ∧ µ β (σ x,± ) .
This identity will be used throughout the paper, without further notice, to replace the left-hand side, which appears in the Dirichlet and in the Thomson principle, by the right-hand side. We turn to an estimate of hitting times in terms of capacities. Fix x ∈ E \(A∪B). Then,

P x [H A < H B ] = P x [ H A < H + B∪{x} ] P x [ H A∪B < H + x ] • (2.11)
Indeed, intersecting the event {H A < H B } with {H + x < H A∪B } and its complement, by the Strong Markov property,

P x [H A < H B ] = P x [ H + x < H A∪B ] P x [H A < H B ] + P x [ H A < H + B∪{x} ] ,
which proves (2.11) by substracting the first term on the right hand side from the left hand side.

Recall the definition of the capacity introduced in (2.8). Multiplying and dividing the right hand side of (2.11) by M (x) yields that

P x [H A < H B ] = M (x) P x [ H A < H + B∪{x} ] cap(x, A ∪ B) ≤ M (x) P x [ H A < H + x ] cap(x, A ∪ B) •
Therefore, by definition of the capacity and since, by (2.9), the capacity is monotone,

P x [H A < H B ] ≤ cap(x, A) cap(x, A ∪ B) ≤ cap(x, A) cap(x, B) • (2.12)
Trace process. We recall in this subsection the definition of the trace of a Markov process on a proper subset of the state space. Fix F E and denote by T F (t) the time the process X t spent on the set F in the time-interval [0, t] :

T F (t) = t 0 χ F (X s ) ds ,
where χ F represents the indicator function of the set F . Denote by S F (t) the generalized inverse of the additive functional T F (t) :

S F (t) = sup{s ≥ 0 : T F (s) ≤ t} .
The recurrence guarantees that for all t > 0, S F (t) is finite almost surely.

Denote by X F (t) the trace of the chain X t on the set F , defined by X F (t) := X(S F (t)). It can be proven [START_REF] Beltrán | Tunneling and metastability of continuous time Markov chains[END_REF] that X F (t) is an irreducible, recurrent, continuoustime, F -valued Markov chain. The jump rates of the chain X F (t), denoted by r F (x, y), are given by

r F (x, y) = λ(x) P x H + F = H y , x , y ∈ F , x = y .
The unique stationary probability measure for the trace chain, denoted by π F , is given by π F (x) = π(x)/π(F ). Moreover, π F is reversible if so is π.

Estimates of an eigenfunction. We derive in this subsection an estimate needed in the next sections. Consider the continuous-time Markov chain X t on E = {0, . . . , n} which jumps from k to k + 1 at rate ε and from k + 1 to k at rate 1, 0 ≤ k < n.

Denote by P k the distribution of the Markov chain X t starting from k ∈ E. Expectation with respect to P k is represented by E k .

Denote by H n the hitting time of n. Fix θ > 0, and let f : E → R + be given by

f (k) = E k e -θ
Hn .

An elementary computation based on the strong Markov property shows that f is the solution of the boundary-valued elliptic problem

(Lf )(k) = θ f (k) , 0 ≤ k < n , f (n) = 1 ,
where L stands for the generator of the Markov chain X t .

Lemma 2.3.1. We have that f (0) ≤ ε n /θ.

Proof. Multiplying the identity (Lf )(k) = θ f (k) by ε k and summing over 0

≤ k < n yields that n-1 k=0 ε k+1 [f (k + 1) -f (k)] + n-1 k=1 ε k [f (k -1) -f (k)] = θ n-1 k=0 f (k) ε k .
On the left-hand side, all terms but one cancel so that

ε n [f (n) -f (n -1)] = θ n-1 k=0 f (k) ε k . Since f (k) ≥ 0 and f (n) = 1, we have that θ f (0) ≤ θ n-1 k=0 f (k) ε k = ε n [1 -f (n -1)] ≤ ε n , as claimed.
This result has a content only in the case ε < 1, but we did not use this condition in the proof.

The emergence of a critical droplet

In this section, we prove that starting from -1, the process creates a droplet of 0-spins on its way to {0, +1}, that is, a configuration σ with a n 0 ×(n 0 +1) rectangle of 0-spins (or 0-rectangle) and an extra 0-spin attached to one of the sides of the rectangle, in a background of negative spins.

In the next section, we prove that if this extra 0-spin is attached to one of the longest sides of the rectangle, with a positive probability the process hits 0 before {-1, +1}, while if it is attached to one of the shortest sides, with probability close to 1, the process returns to -1 before hitting {0, +1}. An important feature of this model is that the size of a critical droplet is independent of β and L.

Throughout this section, C 0 is a large constant, which does not depend on β or L but only on h, and whose value may change from line to line.

Recall the definition of the valley V -1 introduced just above equation (2.4). Note that there are few configurations in V -1 which differ from -1 at n 0 (n 0 + 1) + 1 sites. Moreover, such configurations may not have two spins equal to +1.

(2.13) Since B + is the boundary of the valley V -1 , starting from -1, it is reached before the chain attains the set {0, +1} :

H B + < H {0,+1} P -1 a.s. (2.15)
Note that all configurations of R a have the same energy and that

H(ξ) = H(ζ) + 2 -h if ξ ∈ R a , ζ ∈ R.
In particular, by Assertion 4.D in [START_REF] Landim | Metastability of the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field[END_REF],

H(σ) ≥ H(ξ) + h , σ ∈ B \ R , ξ ∈ R a .
(2.16)

On the other hand, for a configuration

η ∈ R + \ R a , H(η) ≥ H(ζ) + 4 -h if ζ ∈ R, so that H(η) ≥ H(ξ) + 2 , η ∈ R + \ R a , ξ ∈ R a .
(2.17)

In particular, at the boundary B + the energy is minimized by configurations in R a . This means that σ t should attained B + at R a . This is the content of the main result of this section. Let

ε(β) = |Λ L | e -2β + e -hβ .
(2.18) Proposition 2.4.1. There exists a finite constant C 0 such that

P -1 [H B + < H R a ] ≤ C 0 ε(β) for all β ≥ C 0 .
Proof. The proof of this lemma is divided in several steps. Denote by {η t : t ≥ 0} the process obtained from the Blume-Capel model by forbiding any jump from the valley V -1 to its complement. This process is sometimes called the reflected process.

It is clear that η t is irreducible and that its stationary state, denoted by µ V is given by µ V (σ) = (1/Z V ) exp{-βH(σ)}, where Z V is a normalizing constant.

Moreover, starting from -1, we may couple σ t with η t in such a way that σ t = η t until they hit the boundary. Hence, if we denote by P V -1 the distribution of η t ,

P -1 [H B + < H R a ] = P V -1 [H B + < H R a ] .
By (2.12),

P V -1 [H B + < H R a ] = P V -1 [H B + \R a < H R a ] ≤ cap V (B + \ R a , -1) cap V (R a , -1) ,
where cap V represents the capacity with respect to the process η t . The lemma now follows from Lemmata 2.4.2 and 2.4.3 below.

Denote by Γ c the energy of a configuration σ ∈ R a :

Γ c = 4 (n 0 + 1) -h n 0 (n 0 + 1) + 1 -|Λ L | . (2.19)
Lemma 2.4.2. There exists a finite constant C 0 such that Proof. We use the Thomson principle to bound this capacity by constructing a flow from -1 to R a . The flow is constructed in two stages.

1 cap V (R a , -1) ≤ C 0 1 |Λ L | Z V e βΓc .
To explain the procedure we interpret a flow as a mass transport, φ(η, ξ) representing the total mas transported from η to ξ. The goal is to define the transport of a mass equal to 1 from -1 to R a . The first step consists in transferring the mass from -1 to R. This is done as follows. Consider the sequence of points in Z 2 which forms a succesion of squares of length 1, 2 up to n 0 . It is given by u 1 = (1, 1), u 2 = (1, 2), u 3 = (2, 2), u 4 = (2, 1), u 5 = (1, 3) and so on until u n 2 0 = (n 0 , 1). Hence, we first add a new line on the upper side of the square from left to right, and then a new column on the right side from top to bottom. Once we arrived at the (n 0 × n 0 )-square, we add a final row at the upper side of the square : Let u n 2 0 +k = (k, n 0 + 1), 1 ≤ k ≤ n 0 , to obtain a n 0 × (n 0 + 1)-rectangle.

Note that we reach through this procedure only rectangles whose height is larger than the length. We could have defined flows which reach both types of rectangles, but the bound would not improve significantly.

Let A k = {u 1 , . . . , u k } and denote by A x,k the set A k translated by x ∈ Z 2 : A x,k = x + A k . Denote by ζ x,k the configuration with 0-spins at A x,k and -1-spins elsewhere. Let u n 2 0 +n 0 +1 = (1, n 0 + 2), and consider the configurations ζ x,n 2 0 +n 0 +1 obtained through the correspondance adopted above. The final stage consists in transferring the mass ε from ζ x,n 2 0 +n 0 to ζ x,n 2 0 +n 0 +1 . Since each configuration ζ x,n 2 0 +n 0 +1 belongs to R a , the total effect of this procedure is to transport a mass equal to 1 from the configuration -1 to the set R a .

Denote this flow by φ, so that φ(-

1, ζ x,1 ) = φ(ζ x,k , ζ x,k+1 ) = ε.
We extend this flow by imposing it to be anti-symmetric and to vanish on the other bonds. It is clear that this flow belongs to F -1,R a , the set of flows defined above 2.10. Therefore, by the Thomson principle,

1 cap V (R a , -1) ≤ ε 2 |Λ L | n 0 (n 0 +1) k=0 1 µ V (ζ 1,k ) ∧ µ V (ζ 1,k+1 ) • Since ε = 1/|Λ L | and µ V (ζ 1,k ) ≥ µ V (ζ 1,n 0 (n 0 +1)+1 ), the previous expression is bounded by 1 cap V (R a , -1) ≤ C 0 1 |Λ L | 1 µ V (ζ 1,n 0 (n 0 +1)+1 ) ,
which completes the proof of the lemma because the energy of the configuration ζ 1,n 0 (n 0 +1)+1 is Γ c .

We turn to the upper bound for cap(B + \ R a , -1).

Lemma 2.4.3. There exists a finite constant C 0 such that

cap V (-1, B + \ R a ) ≤ C 0 1 Z V |Λ L | e -βΓc |Λ L | e -2β + e -hβ .
for all β ≥ C 0 .

The proof of this lemma is divided in several steps. Let B := B + \ R a and let χ B : Ω L → R be the indicator of the set B. Since χ B (-1) = 0 and χ B (σ) = 1 for σ ∈ B, by the Dirichlet principle,

cap V (B + \ R a , -1) ≤ D V (χ B ) , ( 2.20)
where D V (f ) stands for the Dirichlet form of f for the reflected process η t . An elementary computation yields that

D V (χ B ) = σ∈B σ ∈V -1 \B µ V (σ) ∧ µ V (σ ) , (2.21)
where the second sum is carried over all configurations σ which belong to V -1 \ B and which differ from σ at exactly one spin. We denote this relation by σ ∼ σ.

Let B 1 := B \ R , B 2 := R + \ R a , ( 2.22)
so that B = B 1 ∪ B 2 , and consider separately the sums over B 1 and B 2 . We start with B 2 .

Assertion 2.4.4. We have that

σ∈B 2 σ ∼σ µ V (σ) ∧ µ V (σ ) ≤ C 0 1 Z V |Λ L | e -βΓc |Λ L | e -2β + e -[10-h]β .
Proof. A configuration η ∈ B 2 has a n 0 × (n 0 + 1) rectangle of 0-spins, and an extraspin. If this extra-spin is attached to the rectangle it is equal to +1, while it may be 0 or +1 if it is not. We study the two cases separately. Fix a configuration η ∈ B 2 where the extra-spin is attached to the rectangle, so that H(η) = Γ c + (10 -h). Consider a configuration σ ∈ V -1 \ B such that σ ∼ η. As σ may have at most n 0 (n 0 + 1) + 1 spins different from -1, this excludes the possibility that σ is obtained from η by flipping a -1. By (2.13), configurations in B with n 0 (n 0 + 1) + 1 spins different from -1 may not have two spins equal to +1. This excludes flipping a 0 to +1. Finally, we may not flip the +1 to 0 because by doing so we obtain a configuration in R a , and thus not in

B = B + \ R a .
Hence, either σ is obtained from η by flipping the +1 to -1, or it is obtained by flipping a 0 to -1. In the first case H(σ ) < H(η), while in the second case if the 0-spin belongs to the corner, H(σ ) > H(η). Since the number of configurations obtained by these flips is bounded by a finite constant, the contribution to the sum appearing in the statement of the assertion is bounded by

C 0 η µ V (η) ≤ C 0 1 Z V e -β Γc |Λ L | e -[10-h]β ,
where the factor |Λ L | comes from the number of possible positions of the rectangle, while the constant C 0 absorbs the number of positions of the positive spin. Fix now a configuration η ∈ B 2 where the extra-spin is not attached to the rectangle. Then H(η) ≥ Γ c + 2. Consider a configuration σ ∈ V -1 \ B such that σ ∼ η. As before, since σ may have at most n 0 (n 0 + 1) + 1 spins different from -1, this excludes the possibility that σ is obtained from η by flipping a -1. By excluding this possibility, we are left with a finite number [depending on n 0 ] of possible jumps. Hence, the contribution of configurations of this type to the sum appearing in the statement of the assertion of is bounded by

C 0 η µ V (η) ≤ C 0 1 Z V e -βΓc |Λ L | 2 e -2β ,
where the factor |Λ L | 2 appeared to take into account the possible positions of the rectangle and of the extra particle. This proves the assertion.

It remains to examine the sum over B 1 . Denote by N (σ) the number of spins of the configuration σ which are different from -1 :

N (σ) = #A(σ) = #{x ∈ Λ L : σ x = -1} . ( 2.23)
Next assertion states that we can restrict our attention to configurations σ which have no spin equal to +1. For a configuration σ such that N (σ) ≤ n 0 (n 0 + 1) + 1, let σ o be the one obtained from σ by replacing all spins equal to +1 by 0-spins :

σ o x = σ x ∧ 0. Assertion 2.4.5. For all σ ∈ Ω L such that N (σ) ≤ n 0 (n 0 + 1) + 1, H(σ o ) ≤ H(σ) .
Proof. This result is clearly not true in general because +1 is the ground state. It holds because we are limiting the number of spins different from -1.

For a configuration σ ∈ Ω L , denote by I a,b (σ), -1 ≤ a < b ≤ 1, the number of unordered pairs {x, y} of Ω L such that x -y = 1, {σ x , σ y } = {a, b}, where z stands for the Euclidean norm of z ∈ R 2 .

An elementary computation yields that

H(σ) -H(σ o ) = I 0,1 (σ) + 3 I -1,1 (σ) -hN 1 (σ) ,
where N 1 (σ) stands for the total number of spins equal to +1 in the configuration σ. To prove the assertion, it is therefore enough to show that hN 1 (σ) ≤ I 0,1 (σ) +

I -1,1 (σ). By [19, Assertion 4.A], I 0,1 (σ) + I -1,1 (σ) ≥ 4 N 1 (σ). It remains to obtain that hN 1 (σ) ≤ 4 N 1 (σ), i.e., that h N 1 (σ) ≤ 4. Indeed, since N (σ) ≤ n 0 (n 0 + 1) + 1, N 1 (σ) ≤ n 0 (n 0 +1)+1 so that, by definition of n 0 , h N 1 (σ) ≤ h(n 0 +1) ≤ 2+h ≤ 3. Recall that A(σ) = {x ∈ Λ L : σ x = -1}. A set A ⊂ A(σ) is said to be a connected component of A(σ) if (a)
for any x, y ∈ A, there exists a path (x 0 = x, x 1 , . . . , x m = y) such that x i ∈ A, x i+1 -x i = 1, 0 ≤ i < m and (b) for any x ∈ A, y ∈ A, such a path does not exists.

Next assertion gives an estimation of the energy of a configuration σ ∈ Ω L such that N (σ) ≤ n 0 (n 0 + 1) + 1 in terms of the number of connected components. Assertion 2.4.6. Let σ ∈ Ω L be a configuration such that N (σ) = n 0 (n 0 + 1), and denote by k, 1 ≤ k ≤ n 0 (n 0 + 1) the number of connected components of σ. Then,

H(σ) ≥ Γ c + 2 (k -1) + h .
Proof. By Assertion 2.4.5, we can assume that σ has no spin equal to +1. For such a configuration and by definition of Γ c ,

H(σ) = I -1,0 (σ) -(4n 0 + 4) + Γ c + h .
To complete the proof of the assertion, we have to show that I -1,0 (σ) ≥ (4n 0 + 4) + 2(k -1).

By moving 2 of the connected components of σ, and gluing them together, we reach a new configuration σ 1 such that N (σ 1 ) = N (σ), while the size of the interface has decreased at least by 2 :

I -1,0 (σ) ≥ I -1,0 (σ 1 ) + 2 .
Iterating this argument k -1 times, we finally reach a configuration σ * with only one connected component and such that I -1,0 (σ) ≥ I -1,0 (σ * ) + 2(k -1) .

(2.24)

The last connected component is glued to the set formed by the previous ones in such a way that the set A(σ * ) is not a n 0 ×(n 0 +1) rectangle. This is always possible. Since the connected set A(σ * ) is not a n 0 × (n 0 + 1) rectangle, by [2, Assertion 4.B], I -1,0 (σ * ) ≥ 4n 0 + 4, so that I -1,0 (σ) ≥ (4n 0 + 4) + 2(k -1), which proves the assertion.

We estimate the sum over σ ∈ B 1 on the right-hand side of (2.21) in the next assertion.

Assertion 2.4.7.

There exists a finite constant C 0 such that

σ∈B 1 σ ∼σ µ V (σ) ∧ µ V (σ ) ≤ C 0 1 Z V |Λ L | e -βΓc e -hβ
for all β ≥ C 0 .

Proof. The proof of this assertion is divided in three steps. The first one consists in applying Assertion 2.4.5 to restrict the first sum to configurations with no spin equal to +1. Indeed, as configuration in B 1 have at most n 0 (n 0 + 1) spins different from -1, by this assertion,

σ∈B 1 σ ∼σ µ V (σ) ∧ µ V (σ ) ≤ 2 n 0 (n 0 +1) σ∈B 1,0 σ ∼σ µ V (σ) ∧ µ V (σ ) ,
where B 1,0 represents the set of configurations in {-1, 0} Λ which belong to B 1 .

The second step consists in characterizing all configurations σ which may appear in the second sum. Recall that it is performed over configurations σ ∈ V -1 \ B which can be obtained from σ by one flip. In particular, N (σ ), introduced in (2.23), can differ from N (σ) = n 0 (n 0 + 1) by at most by 1.

If N (σ ) = N (σ) + 1, as σ B ⊃ R + , we have that σ ∈ R a . Since σ does not belong to R, the 0-spins of the configuration σ form a n 0 ×(n 0 +1)-rectangle in which one site has been removed and one site at the boundary of the rectangle has been added. In this case H(σ) > H(σ ) and H(σ) ≥ Γ c + h. The last bound is attained if the site removed from the rectangle to form σ is a corner. Hence, restricting the second sum to configurations σ such that N (σ ) = N (σ) + 1 yields that

σ∈B 1,0 σ ∼σ µ V (σ) ∧ µ V (σ ) ≤ C 0 1 Z V |Λ L | e -βΓc e -hβ ,
where the factor |Λ L | takes into account the possible positions of the rectangle and the constant C 0 the positions of the erased and added sites.

If N (σ ) = N (σ), resp. N (σ ) = N (σ) -1, as σ has no spin equal to +1, this means that one 0-spin has been flipped to +1, resp. to -1. In both cases, there are n 0 (n 0 + 1) such configurations σ . Hence, the sum restricted to such configurations σ is less than or equal to

C 0 σ∈B 1,0 µ V (σ) .
Let N := n 0 (n 0 + 1), and denote by C k , 1 ≤ k ≤ N , the set of configurations in B 1,0 which have k connected components. Rewrite the previous sum according to the number of components and apply Assertion 2.4.6 to obtain that it is bounded by

N k=1 σ∈C k µ V (σ) ≤ C 0 1 Z V |Λ L | e -βΓc e -hβ N k=1 |Λ L | k-1 e -β[2(k-1)] ,
where |Λ L | k takes into account the number of positions of the k components, and C 0 the form of each component. By the assumption of the theorems, |Λ L | e -2β is bounded by 1/2 for β large enough, so that the sum is bounded by 2. To complete the proof of the assertion it remains to recollect the previous estimates.

Proof.[Proof of Lemma 2.4.3] This Lemma is a consequence of Assertions 2.4.4, 2.4.7, and from the fact that h < 5.

(B1) To increase in any direction a rectangle with 2n 0 adjacent 0-spins whose distance from the corners is larger than 2n 0 to a rectangle of adjacent 0-spins which is at distance less than 2n 0 from one of the corners ; (B2) To increase a rectangle with k ≥ 2n 0 adjacent 0-spins which contains one corner and is at a distance larger than 2n 0 from the other corner to a rectangle of adjacent 0-spins which is at distance less than 2n 0 from this later corner. To avoid a too strong assumption on the rate at which the cube Λ L increases, we do not impose [as in the Assertions 2.5.2-2.5.3 and Lemma 2.5.4] the extra rectangle of 0-spins to grow without never shrinking or to grow while the spins at the corners stay put.

As in the proof of Lemma 2.5.7, we construct a set of configurations in two stages. We consider below the case in which the extra rectangle is far from the corners. The case in which it contains one of the corners can be handled similarly.

Fix n 0 ≤ m ≤ n ≤ L -3. Denote by σ a configuration in which nm 0-spins form a (n × m)-rectangle in a sea of -1-spins. Denote this rectangle by A(σ), and assume, without loss of generality, that m is the length and n the height of A(σ). Let (x, y) be the position of the upper-left corner of A(σ).

We attach to one of the sides of A(σ) an extra (p × 1)-rectangle of 0-spins, where p > n 0 . In particular, the length of the side to which this extra rectangle is attached has to be larger than n 0 . To fix ideas, suppose that the extra 0-spins are attached to the upper side of length m of the rectangle and assume that m > 5n 0 . As explained previously, the case m ≤ 5n 0 is handled by Lemma 2.5.4. c,d) by sequentially flipping to -1, close to the corners of A(σ), a total of k 0-spins surrounded, at the moment they are switched, by two -1 spins. We do not flip spins in R c,d .

In the case [not considered below] where the rectangle R c,d includes one corner, say c = 0, we treat the spins at (x, y + 1), . . . , (x + n 0 , y + 1) as belonging to the corner and we allow them to be flipped.

Let

G = ∪ c,d ∪ 0≤k<n 0 G c,d,k
, where the first union is performed over all indices such that 2n 0 ≤ c < d ≤ m -2n 0 , d -c > n 0 . Note that we excluded k = n 0 in this union. Denote by B the configurations which do not belong to G and which can be obtained from a configuration in G by flipping one spin. The set B is treated as the boundary of G.

Consider three independent Markov chains, X t , Y t , Z t . The first one takes value in {0, . . . , m}, it starts from 2n 0 , and jumps from k to k + 1 at rate 2 and from k + 1 to k at rate 2e -βh . The process Y t takes value in {0, . . . , n 0 }, it starts from 0, and jumps from k to k + 1 at rate 2e -βh and from k + 1 to k at rate 1. The last one takes value in {0, 1}, it starts from 0, and jumps from 0 to 1 at rate δ 4 (β).

Before time H B we may couple (x t , |C t |, b t ) with (X t , Y t , Z t ) in such a way that

X t = x t , |C t | ≤ Y t and b t ≤ Z t . In particular, {τ 0 < τ * } ⊂ {H X 0 < H X m }, {H C n 0 < τ * } ⊂ {H Y n 0 < H X m }, {H b 1 < τ * } ⊂ {H Z 1 < H X m }.
In these formulas, H W stands for the hitting time of the process W . Hence, the probability appearing in the statement of the lemma is bounded above by

P H X n 0 < H X m + P H Y n 0 < H X m < H X n 0 + P H Z 1 < H X m < H X n 0 . ( 2

.31)

We estimate each term separately. The first one is easy. Denote by P X k the distribution of

X t starting from k. Let f (k) = P X k [H X n 0 < H X m ]
, which is harmonic. It can be computed explicitly and one gets that

P H X n 0 < H X m = P X 2n 0 H X n 0 < H X m ≤ 2e -n 0 hβ provided β ≥ C 0 .
We turn to the second term of (2.31). On the set {H X m < H X n 0 } we may replace X by a random walk on Z and estimate P H

Y n 0 < H X m ].
P -1 [H +1 < H 0 ] ≤ (2/3) P -1 [H +1 < H 0 ] σ∈R a q(σ) + C 0 δ(β) ≤ (2/3) P -1 [H +1 < H 0 ] + C 0 δ(β) .
This completes the proof of the theorem since δ(β) → 0.

Proof.[Proof of Theorem 2.2.1] Since the chains hits B + before reaching 0 and R l , by the strong Markov property,

P -1 [H R l < H 0 ] = σ∈B + P -1 [H σ = H B + ] P σ [H R l < H 0 ] .
Recall the definition of q(σ) introduced in the previous proof. By Proposition 2.4.1, this expression is equal to σ∈R l q(σ) + σ∈R s q(σ) P σ [H R l < H 0 ] + R(β) .

(2.34)

where the absolute value of the remainder R(β) is bounded by C 0 ε(β). By Assertion 2.5.2, Lemma 2.5.4 and by the proof Lemma 2.5.5, uniformly in σ ∈ R s , σ ∈ R

P σ [H R < H R l ∪{-1,0} ] ≥ 1 -C 0 [δ 1 (β) + δ 2 (β)] , P σ [H -1 < H R l ∪{0} ] = 1 -C 0 δ 2 (β) .
Hence, uniformly in σ ∈ R s ,

P σ [H -1 < H R l ∪{0} ] ≥ 1 -C 0 [δ 1 (β) + δ 2 (β)] , ( 2
β | ≤ C 0 [ε(β) + δ(β)]
for some finite constant C 0 . By Proposition 2.5.1, starting from σ ∈ R a we reach {-1, 0} before +1 with a probability close to 1. Hence, up to a small error, we may include the event H {-1,0} < H +1 inside the second probability which becomes {H 0 < H +1 < H -1 }. Applying the strong Markov property, the right-hand side of the previous expression becomes 4(2n 0 + 1) P 0 [H +1 < H -1 ] We claim that cap(-1, +1)/cap(0, +1) → 0. By Lemma 2.7.5, we may replace the numerator by cap(-1, {0, +1}), and by the second identity of (2.38), we may replace the denominator by cap(0, {-1, +1}). At this point, the claim follows from Proposition 2.7.3. Therefore, lim β→∞ cap(+1, {-1, 0}) cap(0, +1) = 1 .

σ∈R a P -1 [H σ = H B + ] P σ [H 0 < H {-1,+1} ] + R
To complete the proof, it remains to recall again the second identity in (2.38).

Fix k < N and consider the set A 0 k,j of configurations in A 0 k for which the 0-spins have j connected components. There are at most C 0 |Λ L | j of such configurations, and the energy of one of them, denoted by σ, is equal to H(-1) -kh + I -1,0 (σ), where I -1,0 (σ) represents the size of the interface. By (2.24), I -1,0 (σ) ≥ I -1,0 (σ * )+2(j -1), where σ * is configuration obtained from σ by gluing the connected components. By A similar result holds for the pairs (0, V 0 ), (+1, V +1 ), where the valleys are defined analogously as V -1 .

Denote by R l 2 , resp, R s 2 , the set of configurations with n 0 (n 0 + 1) + 2 0-spins in a background of -1-spins. The 0-spins form a [n 0 × (n 0 + 1)]-rectangle with two extra contiguous 0-spins attached to one of the longest, resp. shortest, sides of the rectangle. ) is a configuration with (n 0 + 1) 2 0-spins forming a square in a sea of -1-spins. The previous expression is thus bounded by

Lemma

t δ 1 (β) + 1 θ β E ξ E c,d + 1 θ β max η E η H 0 ∧ tθ β .
Compare the previous expression with the one on the statement of the lemma. We replaced the set R l 2 by the set of (n 0 + 1) 0-squares at the cost of the error term t δ 1 (β), and the expectation of the hitting time of the boundary of the neighborhood of ξ. Proceeding in this way until hitting 0 will bring the sum of all errors and the sum of the expectations of all hitting times. The assumptions of Theorem 2.2.2 where inserted to guarantee that the sum of the error terms converge to 0 as β → ∞. The hitting times of the boundaries involve either the creation of two contiguous 0-spin at the boundary of a rectangle, whose order is e (2-h)β , or the filling of a side of a rectangle, which corresponds to the hitting time of an asymmetric one-dimensional random walk, whose order is proportional to the length of the rectangle. Both orders are much smaller than θ β , which completes the proof of the lemma.

  A} représente le temps d'atteinte d'une configuration σ, Γ c = 4(n 0 + 1) -h[(n 0 + 1)n 0 + 1] est l'énergie nécessaire à la formation d'une goutte critique et n 0 = [2/h], où [•] désigne la partie entière.

  (b) There is an extra 0 spin attached to the longest side of the rectangle ; (c) All the remaining |Λ L | -[n 0 (n 0 + 1) + 1] spins are equal to -1. The configuration σ in Figure 1.1 is an example of a configuration in R l with n 0 = 5.

  Denote by Γ A,B the set of self-avoiding paths from A to B and let H(A, B) := min γ∈Γ A,B H(γ) , H(γ) := max 0≤i≤n H(ξ i ) .

  rectangle. Since all spins of σ are either -1 or 0. By definition of the energy, H(σ) -H(ζ) = P (A(σ)) -P (A(ζ)), and the result follows from Assertion 1.4.2.

Figure 1 . 1 -

 11 Figure 1.1 -Examples of configurations σ ∈ R c and σ ∈ R i in the case where n 0 = 5. A 1 × 1 square centered at x has been placed at each site x occupied by a 0-spin. All the other spins are equal to -1.

σ∈R a 0 PAssertion 1 . 5 . 4 .

 0154 Assertion 1.5.1, the right hand side is equal to lim β→∞ σ [H {-1,+1} < H 0 ] . By Assertion 1.5.2, this expression is equal to (1/2)|R lc 0 | + (2/3)|R li 0 | = 2|Λ L |{2 + (4/3)(n 0 -1)}, which completes the proof of the second claim of the proposition. As in Assertion 1.4.11 we have that lim β→∞ cap(0, +1) cap(0, {-1, +1}) = 1 . (1.30) We have that lim β→∞ cap(-1, +1) cap(-1, {0, +1}) = 1 .

Assertion 1 . 5 . 5 . 1 .

 1551 3, this limit is equal to (1/2)|R lc | + (2/3)|R li |, which completes the proof of the assertion. We have that lim β→∞ cap(+1, {-1, 0}) cap(0, {-1, +1}) = Proof. Indeed, by monotonicity of the capacity and by (1.14), cap(+1, 0) ≤ cap(+1, {-1, 0}) ≤ cap(+1, 0) + cap(+1, -1) . By Assertion 1.5.4, by (1.30), and by Proposition 1.2.3, cap(+1, -1)/cap(0, +1) → 0 as β ↑ ∞.

  3.3 are in force for E 1 = M. Recall Remark 1.3.4. Assertion 1.5.6. Consider two configurations σ ∈ M and η

Lemma 1 . 7 . 4 .

 174 As β ↑ ∞, the speeded-up chain η(θ β t) converges to the continuoustime Markov chain on M in which +1 is an absorbing state, and whose jump rates r(η, ξ), are given by r(-1, 0) = r(0, +1) = 1 , r(-1, +1) = r(0, -1) = 0 . Proof. Denote by r β (η, ξ) the jump rates of the chain η(θ β t). It is enough to prove that lim β→∞ r β (η, ξ) = r(η, ξ) (1.41) for all η = ξ ∈ M. By [2, Proposition 6.1], the jump rates r β (η, ξ), η = ξ ∈ M, of the Markov chain η β (t) are given by r β (η, ξ) = θ β λ(η) P η [H ξ = H + M ] . Dividing and multiplying the previous expression by P η [H M\{η} < H + η ], in view of [2, Lemma 6.6] and of (1.10), we obtain that

Lemma 1 . 7 . 6 .

 176 Let ∆ = Ω \ V. For all ξ ∈ V, t > 0β ) ∈ ∆} ds = 0 .

Lemma 1 . 7 . 7 .

 177 β ) ∈ ∆} ds = 0 , for ζ = 0 (resp. ζ = -1), which follows from Lemma 1.7.7 below. Let ∆ * = Ω \ M. For all ξ ∈ M, t > 0, β ) ∈ ∆ * } ds = 0 . Proof. Consider first the case ξ = +1. Clearly,

  we may add in the previous expectation the indicator of the set {H +1 < H -1 }. Rewrite the integral over the time interval [0, tθ β ] as the sum of an integral over [0, H {-1,+1} ] with one over the time interval [H {-1,+1} , tθ β ]. The expectation of the first one is handled by (1.47). The expectation of the second one, by the strong Markov property, on the set {H {-1,+1} ≤ tθ β } ∩ {H +1 < H -1 }, is less than or equal to

Figure 2 . 3 -

 23 Figure 2.3 -We present in this figure some configurations ζ x,k introduced in the proof of Lemma 2.4.2. Let m = 16. The figure represent the configurations ζ x,m , ζ x,m+1 , ζ x,m+2 . Then, ζ x,m+4 , ζ x,m+5 , ζ x,m+6 , and ζ x,m+9 . An 1 × 1 empty square centered at x has been placed at each site x occupied by a 0-spin. All the other spins are equal to -1.

Figure 2 .

 2 3 presents some of these configurations. Let ε = 1/|Λ L |. The first stage of the flow consists in transferring a mass ε from -1 to each ζ x,1 and then transfer this mass from ζ x,k to ζ x,k+1 for 1 ≤ k < n 0 (n 0 + 1).

Figure 2 . 4 -

 24 Figure 2.4 -Assume that n 0 = 3. The first picture provides an example of a configuration η(c,d) . Here, m = 18 ≤ n, c = 6, d = 10 and p = 5. The gray portion indicates that the rectangle continues below as its height is larger than 18. The second picture presents a configuration in G c,d,6 . We chose k = 6 > n 0 to make the definition clear.

0 ≤ 1 ≤ 3 δ 4 Remark 2 . 5 . 9 .

 0134259 As X and Y are independent, we condition on Y and treat H Y n 0 as a positive real number. The set{H Y n 0 < H X m } is contained in {X J ≤ m} where J = H Y n 0 . Fix θ > 0.By the exponential Chebyshev inequality and since m ≤ n [the sizes of the rectangle A(σ)],P X 2n 0 X J ≤ m ≤ P X 0 X J ≤ n ≤ e θn E X 0 e -θX J .Choose θ = 1/n and compute the expectation to obtain that the previous expression is bounded by3 exp{-(2/n)H Y n 0 } provided β ≥ C 0 . By Lemma 2.3.1, E e -(2/n)H Y n C 0 n e -n 0 hβ ,where E represents the expectation with respect to P . The third expression in (2.31) is estimated similarly. The argument yields that it is bounded by3 E e -(2/n)H Z (β) (2/n) + δ 4 (β) ≤ 2 n δ 4 (β) ,where δ 4 (β) has been introduced just above (2.30). This completes the proof of the lemma because n 0 h > 2 -h. One could improve the previous argument and obtain a better estimate by allowing the spins at the boundary of A(σ) to flip while the rectangle R c,d fills the upper side.The next result describes how the supercritical droplet of 0-spins grows. Let δ 5 (β) := e -[(n 0 +1)h-2]β + e -hβ + |Λ L | 3/2 e -(

β 3 • 3 . 2 . 7 . 6 . 1 .

 332761 whose absolute value is bounded byC 0 [ε(β) + δ(β)]. By Theorem 2.2.2, P 0 [H +1 < H -1 ]converges to 1. On the other hand, the sum can be handled as in the proof of Proposition 2.7.3 to yield thatlim β→∞ cap(-1, +1) µ β (σ * ) |Λ L | = 4(2n 0 + 1)This completes the proof of the lemma in view of Proposition 2.7.LemmaWe have thatlim β→∞ cap(+1, {-1, 0}) cap(0, {-1, +1}) =Proof. By monotonicity of the capacity and by equation (3.5) in[START_REF] Landim | Metastability of the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field[END_REF], cap(+1, 0) ≤ cap(+1, {-1, 0}) ≤ cap(+1, 0) + cap(+1, -1) .

[ 2 ,|Λ

 2 Assertion 4.A], I -1,0 (σ * ) ≥ 4 √ k. Therefore, the previous sum for k < N is bounded above by L | j e -2(j-1)β .Since |Λ L | e -2β ≤ 1/2 for β sufficiently large and since 4√ k -kh ≥ min{4h, 4 √ N -1 -(N -1)h}, the previous sum is less than or equal toC 0 |Λ L | e -[4-h]β + e -2(n 0 -1)β ≤ C 0 |Λ L | e -[4-h]β + e -2β ≤ C 0 |Λ L | e -2β because 4 √ N -1 -(N -1)h = 4 n 0 (n 0 + 1) -n 0 (n 0 + 1)h ≥ 4n 0 -2(n 0 + 1) = 2(n 0 -1) ≥ 2.It remains to consider the contribution of the set A N . There are at most C 0 |Λ L | configurations in this set, and each configuration has the same energy. The contribution of these terms to the sum (2.43) is bounded byC 0 |Λ L |e -(2n 0 +1)β ≤ C 0 |Λ L |e -2β .This completes the proof of the lemma.

Assertion 2 . 8 . 3 .

 283 We have that lim ∈ ∆} ds = 0 .Proof. As the process σ t is stopped at time H B + , we may replace ∆ by ∆ ∩ V -1 . By [2, Proposition 6.10], and by definition of θ β , the expression appearing in the statement of the lemma is bounded above by1 θ β µ β (∆ ∩ V -1 ) cap(-1, B + ) = cap(-1, M \ {-1}) cap(-1, B + ) µ β (∆ ∩ V -1 ) µ β (-1) • By Proposition 2.7.3 and (2.36), the first ratio converges to 1/3, while by Lemma 2.8.2 the second one converges to 0.

Lemma 2 . 8 . 7 .θ β tθ β 0 1 µ

 2871 For every t > 0, ∈ ∆} ds = 0 . Proof. By (2.42), we may insert the indicator of the set {H B + +1 ≥ t θ β } inside the expectation. After this insertion, we may replace the process σ t by the reflected process at V +1 , denoted by η t , and then bounded the expression +1 η(s)∈ ∆ ds ≤ 1 V (+1) ξ∈V +1 µ V (ξ) P V ξ η(s) ∈ ∆ ds .

  Beltrán et Landim puisque, bien que la preuve d'un tel résultat soit évidemment divisée en étapes intermédiaires, la plupart d'entre elles concernent l'estimation de capacités. Or, dans le cas où le tore est de taille fixée, l'espace d'états est fixe. L'estimation de la capacité entre des ensembles A et B se rapporte alors à la recherche du chemin allant de A vers B avec la plus faible différence d'énergie, parfois appelée hauteur dans la littérature.Le second résultat principal donne une estimation du temps d'atteinte. On démontre que, partant de -1, le temps d'atteinte de l'état +1 est d'ordre exponentiel :

  1.2.3.A, replace cap(-1, {0, +1}) by the numerator appearing in the statement of this assertion. The proof is identical up to formula (1.29).Proof.[Proof of Proposition 1.2.4]We first assume that the chain starts from 0. By Lemma 1.3.3 and Assertion 1.5.6,

	It remains to estimate
	lim β→∞	σ∈R a	P

σ [H 0 < H {-1,+1} ] which is equal to lim β→∞ σ∈R a P σ [H {0,+1} < H -1 ]

in view of (1.27). This expression has been computed at the end of the proof of Proposition 1.2.3.A, which completes the proof of the assertion.

  The second one is bounded by P σ [H +1 < H 0 ], which vanishes by Proposition 1.2.1. Since H < min{H 0 , H +1 }, by the strong Markov property, the first term is less than or equal to max

	σ ∈V +1

  By the first estimate in(2.33), uniformly in σ ∈ R a ,P σ [H +1 < H 0 ] ≤ P σ [H +1 < H 0 , H {-1,0} < H +1 ] + C 0 δ(β) = P σ [H -1 < H +1 < H 0 ] + C 0 δ(β) .Therefore, by the strong Markov property at time H -1 ,P -1 [H +1 < H 0 ] ≤ P -1 [H +1 < H 0 ] P σ [H -1 < H {0,+1} ] + C 0 δ(β) . because ε(β) ≤ δ(β).By the second bound in (2.33), as δ(β) → 0, for σ ∈ R l , P σ [H -1 < H {0,+1} ] ≤ 2/3 provided β ≥ C 0 . Hence, for β large enough,

	q(σ)
	σ∈R a

2-h)

β .

  .35) and we may introduce the set {H -1 < H R l ∪{0} } inside the probability appearing in (2.34) by paying a cost bounded byC 0 [δ 1 (β) + δ 2 (β)].Up to this point, we proved thatP -1 [H R l < H 0 ] = Proof. Fix a configuration σ * in R a . By the proof of Proposition 2.7.3, cap(-1, +1) µ β (σ * ) |Λ L | = 4(2n 0 + 1) σ∈R a P -1 [H σ = H B + ] P σ [H +1 < H -1 ] + R

	q(σ) + lim β→∞ cap(-1, +1) cap(-1, {0, +1}) q(σ) P Lemma 2.7.5. We have that where |R (1)	= 1 .	(1) β ,
	σ∈R l	σ∈R s

σ [H -1 < H R l < H 0 ] + R(β) ,

where the absolute value of the remainder R(β) is bounded by

C 0 [ε(β) + δ 1 (β) + δ 2 (β)]

. By the strong Markov property this expression is equal to

σ∈R l q(σ) + P -1 [H R l < H 0 ] σ∈R s q(σ) P σ [H -1 < H R l ∪{0} ] + R(β) .

  2.8.4. For every t > 0, ∈ ∆} ds 1 H B + ≥ tθ β = 0 .We may therefore insert the indicator of the set {H B + < tθ β } inside the expectation appearing in the statement of the assertion at a negligible cost. By Proposition 2.4.1, we may also insert the indicator of the set {H B + = H R a }. After inserting these indicator functions, by the strong Markov property, we get that the expectation appearing in the statement of the lemma is bounded by1 θ β E -1 1 H R a = H B + ≤ tθ β E σ(H B + )Since λ β (σ) ≥ 1 for σ ∈ R a and since R a ⊂ ∆, by removing from the integral the interval [0, τ 1 ], where τ 1 represents the time of the first jump, the previous expectation is less than or equal to1 θ β σ∈R a ξ P -1 H σ = H B + p β (σ, ξ) E ξBy Assertions 2.5.2 and 2.5.3, we may disregard all configurations ξ which do not have two contiguous attached 0-spins and which do not belong to R. The previous expression is thus bounded above by∈ ∆} ds + R β ,where R s 2 represents the set of configurations with two contiguous 0-spins attached to the shortest side of the rectangle, and R β → 0. By Lemma 2.8.4, the first term vanishes as β → ∞. The second one can be absorbed in the left-hand side of the expression appearing in the statement of the lemma, which completes the proof of the lemma. E ξ H 0 ∧ tθ β = 0 .Proof. The proof of this lemma follows the steps of Section 2.5 where we described the growth of the supercritical droplet. Fix a configuration ξ in R l 2 and recall Lemma 2.5.4. By this result,1 θ β E ξ H 0 ∧ tθ β ≤ 1 θ β E ξ 1{E c,d = H c,d } (H 0 ∧ tθ β ) + t δ 1 (β) . Since E c,d < H 0 ,we may write H 0 as E c,d + H 0 • E c,d and bound H 0 ∧ tθ β by E c,d + [(H 0 • E c,d ) ∧ tθ β ]. Hence, by the strong Markov property, the previous expression is less than or equal to

	lim β→∞ 1{σ(s) ∈ ∆} ds ≤ 3 lim sup 1 θ β max ξ∈R s 2 ∪R β→∞ Proof. By Assertion 2.8.3, lim sup β→∞ 1 θ β E -1 tθ β 0 0 where R β → 0. The previous expectation is bounded by 1 θ β σ∈R tθ β max ξ∈R l 2 1{σ(s) ∈ ∆} ds 1 θ β E ξ tθ β 0 0 1{σ(s) ∈ ∆} ds + 1{σ(s) ∈ ∆} ds . + R β , 1 θ β • max ξ∈R l 2 1 θ β E ξ tθ β 0 1{σ(s) ∈ ∆} ds + 2 3 1 θ β max ξ∈R s 2 ∪R E ξ tθ β 0 1{σ(s) The second expectations is bounded by 1 θ β max ξ∈R s 2 ∪R E ξ H -1 ∧ tθ β + 2 3 1 θ β E -1 tθ β 0 1{σ(s) ∈ ∆} ds . Lemma 2.8.6. For every t > 0, lim sup β→∞ max ξ∈R l 2 1 θ β E Lemma 2.8.5. For every t > 0, lim β→∞ tθ β 1 θ β E -1 0 1{σ(s) tθ β t δ 1 (β) + 1 θ β 1 E ξ E c,d + θ β E

ξ H -1 ∧ tθ β = 0 . a P -1 H σ = H B + E σ tθ β 0 1{σ(s) ∈ ∆} ds . ξ 1{E c,d = H c,d } E σ(E c,d ) H 0 ∧ tθ β .

On the set

{E c,d = H c,d }, σ(E c,d

Remerciements

78 Bibliographie

To define the boundary of the valley of V -1 , fix L large, and denote by B the set of configurations with n 0 (n 0 + 1) spins different from -1 :

Denote by R the subset of B given by

Note that the spins of a configuration σ ∈ R are either -1 or 0 and that all configurations in R have the same energy.

Denote by R + the set of configurations in Ω L in which there are n 0 (n 0 + 1) + 1 spins which are not equal to -1. Of these spins, n 0 (n 0 + 1) form a n 0 × (n 0 + 1)rectangle of 0 spins. The remaining spin not equal to -1 is either 0 or +1. Let B + be the boundary of V -1 . This set consists of all configurations σ in V -1 which have a neighbor [that is, a configuration σ which differs from σ at one site] which does not belong to V -1 . By definition of V -1 ,

Figure 2.2 -Example of three configurations in R a in the case where n 0 = 5. An 1×1 empty square centered at x has been placed at each site x occupied by a 0-spin. All the other spins are equal to -1. The one on the left belongs to R s . According to the notation introduced at the beginning of Section 2.5, the one on the center belongs to R li and the one on the right to R lc .

Let R a ⊂ R + be the set of configurations for which the remaining spin is a 0 spin attached to one of the sides of the rectangle. Figure 2.2 present some examples of configurations in R a . We write the boundary B + as

(2.14)

The growth of a supercritical droplet

In the previous section we have seen that starting from -1 we hit the boundary of the valley V -1 at R a . In this section we show that starting from R a the process either returns to -1, if the extra 0-spin is attached to one of the shortest sides of the rectangle, or it invades the all space with positive probability, if the extra 0-spin is attached to one of the longest sides of the rectangle.

Denote by R l , R s the configurations of R a in which the extra particle is attached to one of the longest, shortest sides of the rectangle, respectively, and by R c the configurations of R a in which the extra particle is attached to one corner of the rectangle. Let

Recall that M = {-1, 0, +1}, and let

(2.25)

Proposition 2.5.1. There exists a finite constant C 0 such that for all σ ∈ R lc , σ ∈ R li , and σ ∈ R s ,

The proof of this proposition is divided in several lemmata. The first result describes what happens when there is a 0-spin attached to the side of a rectangle of 0-spins in a sea of -1-spins. From such a configuration, either the attached 0-spin is flipped to -1 or an extra 0-spin is created at the neighborood of the attached 0-spin.

For n ≥ 1, let

Proof. We prove the lemma for σ -, the proof for σ + being identical. Clearly,

Consider the last sum. There are three terms, corresponding to the corners of the rectangle, for which R β (σ, σ ) ≤ e -βh . There are 4(n + m) -2 terms, corresponding to the inner and outer boundaries of the rectangle, such that R β (σ, σ ) ≤ e -β (2-h) . All the remaining rates are bounded by e -β(4-h) . Hence,

where κ n (β) has been introduced in (2.26). This proves the assertion.

In the next assertion we consider the case in which the extra 0-spin does not sit at the corner of the rectangle, but in its interior. The proof of this result, as well as the one of the next assertion, is similar to the previous proof.

Assertion 2.5.3. Under the same hypotheses of the previous assertion, assume now that the extra 0-spin has one neighbor 0-spin which does not sit at one corner of the rectangle [there are exactly two -1-spins with two 0-spins as neighbors ]. Let σ -, resp. σ + + , σ - + , be the configuration obtained from σ by flipping to -1 the attached 0-spin, resp. by flipping to 0 one of the two -1-spins with two 0-spins as neighbors. Then, there exists a constant C 0 such that

The next lemma states that once there are two adjacent 0-spins attached to one of the sides of the rectangle, this additional rectangle increases with very high probability. This result will permit to enlarge a (p × 1)-rectangle to a (2n 0 × 1)rectangle. To enlarge it further we will apply Lemma 2.5.8 below. This result will be used in three different situations : (A1) To increase in any direction a rectangle with 2 adjacent 0-spins whose distance from the corners is larger than 2n 0 to a rectangle with 2n 0 adjacent 0-spins ; (A2) To increase in the direction of the corner a rectangle with k ≥ 2 adjacent 0-spins which is at distance n 0 or less than from one of the corners to a rectangle with adjacent 0-spins which goes up to the corner ; (A3) To increase a rectangle with k < 2n 0 adjacent 0-spins which includes one of the corners to a rectangle with 2n 0 adjacent 0-spins. Fix n 0 ≤ m ≤ n ≤ L -3. Denote by σ a configuration in which nm 0-spins form a (m × n)-rectangle in a sea of -1-spins. Recall that we denote this rectangle by A(σ), and assume, without loss of generality, that m is the length and n the height of A(σ). Let (x, y) be the position of the upper-left corner of A(σ).

We attach to one of the sides of A(σ) and extra (p × 1)-rectangle of 0-spins, where p ≥ 2. To fix ideas, suppose that the extra 0-spins are attached to the upper side of length m of the rectangle.

Proof. Consider a configuration η (a,b) in S c,d . To fix ideas assume that a > 0, b < m.

At rate 1 the -1-spins at (x + a -1, y + 1), (x + b + 1, y + 1) flip to 0. Consider all other possible spin flips. There are less than 2 |Λ L | flips whose rates are bounded by e -[4-h]β , 4(n + m) flips whose rates are bounded by e -[2-h]β and 4 flips whose rates are bounded by e -hβ . Since all these jumps are independent, the probability that the -1-spin at (x + a -1, y + 1) flips to 0 before anything else happens is bounded below by 1

. Iterating this argument n 0 -times yields the lemma.

Applying Assertion 2.5.2 or 2.5.3 and then Lemma 2.5.4 to a configuration σ ∈ R a yields that either the process returns to R or an extra row or line of 0-spins is added to the rectangle of 0-spins. The next two lemmata describe how the process evolves after reaching such a configuration.

Denote by m ≤ n the length of the rectangle of 0-spins. If the shortest side has length n 0 or less, the configuration evolves to a (m × [n -1]) rectangle of of 0-spins. If both sides are supercritical, that is if m > n 0 , a -1-spin next to the rectangle is flipped to 0.

Denote by S L the set of stable configurations of Ω L , i.e., the ones which are local minima of the energy :

Consider a configuration σ with nm 0-spins forming a (n × m)-rectangle, all the other ones being -1. If m = n = 2, let S(σ) = {-1}. If this is not the case, let S(σ) be the pair (quaternion if m = n) of configurations in which a row or a column of 0-spins of length m is removed from the rectangle A(σ).

We define the valley of σ, denoted by V σ , as follows. Let G k , 0 ≤ k ≤ m, be the configurations which can be obtained from σ by flipping to -1 a total of k 0-spins surrounded, at the moment they are switched, by two -1-spins. In particular, the elements of G 1 are the four configurations obtained by flipping to -1 a corner of A(σ).

Let G = ∪ 0≤k<m G k . Note that we do not include G m in this union. Let B be the configurations which do not belong to G, but which can be obtained from a configuration in G by flipping one spin. Clearly, S(σ) and G m are contained in B. Finally, let V(σ) = G ∪ B be the neighborhood of σ.

Consider a configuration σ with nm 0-spins forming a (n × m)-rectangle, all the other ones being -1. Then, there exists a constant C 0 such that

Proof. Assume that 2 < m < n. The other cases are treated in a similar way. As in the proof of Proposition 2.4.1, denote by η t the process σ t reflected at V σ , and by P V σ its distribution starting from σ. By (2.12),

where cap V represents the capacity with respect to the process η t . We estimate separately these two capacities. Let η (k) , 0 ≤ k ≤ m, be a sequence of configurations such that η (0) = σ, η (m) ∈ S(σ), and η (k+1) is obtained from η (k) by flipping to -1 a 0-spin surrounded by two -1-spins.

Consider the flow ϕ from σ to S(σ) given by ϕ(η (k) , η (k+1) ) = 1 and ϕ = 0 for all the other bonds. By Thomson's principle,

To estimate the capacity on the numerator, denote by χ = χ B\S(σ) the indicator function of the set B \ S(σ). By the Dirichlet principle,

where the last sum is performed over all configurations σ ∈ V σ \ [B \ S(σ)] which can be obtained from σ by one flip.

To estimate the last sum we examine all elements of B \ S(σ). There are at most C 0 |Λ L | configurations σ obtained from a configuration in G by flipping a spin at distance 2 or more from the [inner or outer] boundary of A(σ). These configurations have only one neighbor σ in V σ and their energy is bounded below by H(σ) + 4 -h.

There are at most C 0 configurations σ not in G m and obtained from a configuration in G by flipping a spin [surrounded by three spins of the same type] at the boundary of A(σ). These configurations have only one neighbor σ in V σ and their energy is bounded below by H(σ) + 2 -h.

Finally there are at most C 0 configurations σ in G m \ S(σ) or obtained from a configuration in G by flipping a spin [surrounded by two spins of the same type] at 57 the boundary of A(σ). These configurations have at most C 0 neighbors σ in V σ and their energy is bounded below by H(σ) + mh. It follows from the previous estimates that

Putting together the previous estimates on the capacity, we conclude that

This completes the proof of the lemma.

Applying the previous result repeatedly yields that starting from a configuration σ with nm 0-spins forming a (n × m)-rectangle in a sea of -1-spins the process converges to -1 if the shortest side has length m ≤ n 0 . Corollary 2.5.6. Let σ be a configuration with n 0 (n 0 + 1) 0-spins which form a n 0 × (n 0 + 1)-rectangle in a background of -1. Then,

The next results shows that, in constrast, if m > n 0 , then the rectangle augments. We first characterize how the process leaves the neighborhood of such a configuration σ.

Fix n 0 < m ≤ n ≤ L -3. Consider a configuration σ with nm 0-spins forming a (n × m)-rectangle in a sea of -1's. Recall that we denote by A(σ) the rectangle of 0-spins. Let V σ be the valley of σ whose elements can be constructed from σ as follows.

Fix 0 ≤ k ≤ n 0 . We first flip sequentially k spins of A(σ) from 0 to -1. At each step we only flip a 0-spin if it is surrounded by two -1-spins. The set of all configurations obtained by such a sequence of k flips is represented by G k . In particular, since at the beginning we may only flip the corners of A(σ), G 1 is composed of the four configurations obtained by flipping to -1 one corner of A(σ). On the other hand, since m > n 0 , all configurations of G k have an energy equal to H(σ) + kh. Denote by G -1 the configuration obtained from σ by flipping to 0 a -1-spin which is surrounded by one 0-spin. Let G = ∪ -1≤k<n 0 G k , and note that G n 0 has not been included in the union.

The second and final stage in the construction of the valley V σ consists in flipping a spin of a configuration in G. More precisely, denote by B all configurations which are not in G, but which can be obtained from a configuration in G by flipping one spin. The set B is interpreted as the boundary of the valley

Note that all configurations in V σ can be obtained from σ by at most n 0 flips. Conversely, if (η (0) = σ, η (1) , . . . , η (n 0 ) ) is a sequence of configurations starting from σ in which each element is obtained from the previous one by flipping a different spin, one of the configurations η (k) belong to the boundary of V σ .

Denote by R 2 the set of 2(m + n -2) configurations obtained from σ by flipping to 0 two adjacent -1-spins, each of which is surrounded by a 0-spin. Clearly, R 2 is contained in B, and the energy of a configuration in R 2 is equal to H -1 = H 0 -h, where H 0 := H(σ)+(2-h) is the configuration in which only one -1-spin has flipped to 0. As n 0 h > 2 -h, an inspection shows that all the elements of A := B \ R 2 have an energy strictly larger that H 0 . In particular, starting from σ, the process reaches the boundary B at R 2 . This is the content of the next lemma.

Let

(2.27)

Then, there exists a constant C 0 such that

Proof. Since we may not leave the set V σ without crossing its boundary B, the probability appearing in the statement of the lemma is equal to the one for the reflected process at V σ , that is, the one in which we forbid jumps from V σ to its complement. We estimate the probability for this later dynamics which is restricted to V σ . By (2.12), the probability appearing in the statement of the lemma is bounded above by cap V (σ, A)/cap V (σ, R 2 ), where cap V stands for the capacity with respect to the reflected process. We estimate the numerator by the Dirichlet principle and the denominator by the Thomson principle.

We start with the denominator. Denote by η (1) , . . . , η (2(n+m-2)) , the configurations of R 2 , and by x j , y j ∈ Z 2 the positions of the two extra 0-spins of η (j) . Assume that

, and ϕ = 0 at all the other bonds. By the Thomson principle, since µ V (σ x j ) is less than or equal to µ V (σ) and µ V (η (j) ),

(2.28)

We turn to the numerator. Denote by f the indicator function of the set A. Since f vanishes at σ and is equal to 1 at A, by the Dirichlet principle, cap V (σ, A) ≤ D V (f ). On the other hand,

where the second sum is performed over all configurations in V σ \ A which can be obtained from η by one spin flip. This relation is represented by ξ ∼ η.

We first consider the configuration η in A which have a neighbor in G -1 . Fix ξ ∈ G -1 . Consider the configurations obtained from ξ by flipping a spin which is not at the boundary of A(σ). There are at most |Λ L | of such spins, and the energy of the configurations obtained by this spin flip is bounded below by H 0 + 4 -h. There is one special spin, though, the one which is next to the extra spin and not at the boundary of A(σ). The energy of the configuration obtained by flipping this spin to 0 or to +1 is bounded below by H 0 + 2 -h. The contribution of these terms to (2.29) is thus bounded above by

where the factor 2 (n + m) comes from the total number of configurations in G -1 .

We turn to the configurations obtained from ξ by flipping a spin at the boundary of A(σ). Since the configuration resulting from this flip can not be in R 2 , their energy is bounded below by H 0 + 2 -h. The contribution of these terms to the sum (2.29) is thus bounded by

the extra factor 2(n + m) coming from the possible positions of the extra spin flip at the boundary. Consider now configurations η in A which have a neighbor in a set G k , 0 ≤ k < n 0 . Fix 0 ≤ k < n 0 and ξ ∈ G k . The configuration ξ is formed by a connected set A(ξ) ⊂ A(σ) of 0-spins in a sea of -1-spins.

There is one special case which is examined separately. Suppose that ξ belongs to G n 0 -1 and η to G n 0 . There are C(n 0 ) of such pairs, and the energy of η is equal to

We exclude from now in the analysis these pairs.

Apart from this case, there are two types of configurations η ∈ A which can be obtained from ξ by a spin flip. The first ones are the ones in which η and ξ differ by a spin which belongs to the inner or outer boundary of A(ξ). There are at most 4(n + m) ≤ 8n of such configurations. The energy of these configurations is bounded below by H(ξ) + 2 -h = H(σ) + kh + 2 -h = H 0 + kh. The minimal case occurs when a -1-spin which has a 0-spin as neighbor is switched to 0.

The previous estimate is not good enough in the case k = 0 because in the argument we did not exclude the configurations in G -1 . For k = 0 if η belongs to B \ G -1 , we obtain that H(η) ≥ H(σ) + 2 + h = H 0 + 2h. The right-hand side of this inequality corresponds to the case in which a 0-spin surrounded by three 0-spins has been changed to -1. In conclusion, if the flip occurs at the boundary of A(ξ), there are at most 8n configurations and the energy of such a configuration is bounded below by H 0 + h.

If the flip did not occur at the boundary of A(ξ), there are at most |Λ L | possible configurations, and the energy of these configurations is bounded below by

The previous estimates yield that the Dirichlet form (2.29) is bounded by

Multiplying this expression by (2.28) yields that the probability appearing in the statement of the lemma is bounded above by

because (n 0 + 1)h -2 < h and 4 -h > 2. We bounded 1/n by 1 when n appeared in the denominator because n can be as small as n 0 + 1. This completes the proof of the lemma since n 2 ≤ |Λ L |.

The previous lemma asserts that the process leaves the neighborhood of a large rectangle of 0-spins in a sea of -1 spins by switching from -1 to 0 two adjacent spins at the outer boundary of the rectangle. At this point, applying Lemma 2.5.4 yields that with a probability close to 1 these two adjacent 0-spins will increase to 2n 0 adjacent 0-spins. To increase it further, we apply the next lemma. This result will be used in two different situations :

Note that B contains configurations in G c,d,n 0 and also configurations in G c,d,k in which d -c = n 0 . Let A 1 , A 2 be such configurations :

where the first union is performed over all indices such that 2n 0 ≤ c < d ≤ m -2n 0 , d -c > n 0 , and the second one is performed over all indices such that 2n 0 < c < d ≤ m -2n 0 , d -c = n 0 . The set B also contains configurations in which a 0-spin in a rectangle R c,d surrounded by 3 0-spins is flipped to ±1.

All configurations in G are similar to the ones represented in Figure 2.4. They are obtained by adding a (p × 1)-rectangle of 0-spins to the upper side of A(σ) and by switching to -1 some of the spins of A(σ) close to the corners.

For t < H B , denote by c t , resp. d t , the position at time t of the leftmost, resp. rightmost, 0-spin of the upper rectangle. Let τ * be the first time

Then, there exists a finite constant C 0 such that

for all β ≥ C 0 .

Proof. Let C t be the set of spins in A(σ) close to the corners which takes the value -1 at time t. In the right picture of Figure 2 Let b t = 1{σ t ∈ G c \ A}. This process starts from 0 and jumps to 1 when σ t reaches B through a configuration which is not in A. Inspecting all possible jumps yields that the process b t is bounded by a process z t which starts from 0 and jumps to 1 at rate

, where δ 4 (β) has been introduced just above (2.30)

The key observation in the proof of this lemma is that the processes (c t , d t ), C t and b t are independent until the set B is attained because they involve different spin jumps.

Let x t = d t -c t . Before hitting B, x t evolves as a random walk in Z which starts from d -c ≥ 2n 0 and jumps from k to k + 1 at rate 2 and from k + 1 to k at rate 2e -βh . Let τ 0 be the first time x t ≤ n 0 .

Let H b 1 be the hitting time of 1 by the process b t , and let H C n 0 be the first time

A simple computation based on the bound |Λ L |e -2β ≤ 1, which holds for β large enough, shows that there exists C 0 such that

for all β ≥ C 0 .

Proposition 2.5.10. Fix n 0 < m ≤ n ≤ L. Let σ be a configuration with mn 0spins which form a m × n-rectangle in a background of -1-spins. Then, there exists a constant C 0 such that

for all β ≥ C 0 . In this equation, if n ≤ L -3, S(σ) is the set of four configurations in which a row or a column of 0 spins is added to the rectangle A(σ). If m < n = L -2, the set S(σ) is a triple which includes a band of 0 spins of width m and two configurations in which a row or a column of 0 spins of length n is added to the rectangle A(σ). If m ≤ L -3, n = L, the set S(σ) is a pair formed by two bands of 0 spins of width m + 1.

Proof. Consider the first case, the proof of the other ones being similar. By Lemma 2.5.7, with a probability close to 1, the process σ t escapes from the valley V σ of σ by flipping to 0 two adjacent spins at the outer boundary of A(σ). By Lemma 2.5.4, with a probability close to 1, these two adjacent spins will become 2n 0 adjacent 0-spins. Of course, if the length of the side is smaller than 2n 0 , this simply means that the 0-spins fill the side. Denote by R e the (2n 0 × 1)-rectangle of adjacent 0-spins. At this point, if R e is at distance less than 2n 0 of one of the corners of A(σ), we apply Lemma 2.5.4 again to extend it up to the corner. After this step, or if R e is at distance greater than 2n 0 of one of the corners of A(σ), we apply Lemma 2.5.8 to increase R e up to the point that one of its extremities is at a distance less than 2n 0 of one of the corners of A(σ). We fill the 2n 0 sites with 0-spins by applying again Lemma 2.5.4. We repeat the procedure applying Lemma 2.5.8 to reach a position close to the corner and then Lemma 2.5.4 to fill the gap.

The probability that something goes wrong in the way is bounded by the sum of the probabilities that each step goes wrong. This is given by C 0 {δ 3 (β) + 6n 0 δ 1 (β) + 2|Λ L | 1/2 δ 4 (β)}, which completes the proof of the proposition in view of (2.32).

Corollary 2.5.11. Let σ be a configuration with n 0 (n 0 + 1) + 2 0-spins which form a n 0 × (n 0 + 1)-rectangle in a background of -1, with two additional adjacent 0-spins attached to the longest side of the rectangle. Then, there exists a constant C 0 such that

Proof. Denote by {T j : j ≥ 1} the jump times of the process σ t . Let σ + be the configuration obtained from σ by flipping to 0 all -1-spins in the same row or column of the two adjacent 0-spins. Hence, σ + has (n 0 + 1) 2 0-spins in a background of -1 spins. According to Lemma 2.5.4, P σ [σ(T n 0 -1 ) = σ + ] ≥ 1 -C 0 δ 1 (β). Since T n 0 -1 ≤ H M , we may apply the strong Markov property to conclude that

At this point, apply Proposition 2.5.10 (2|Λ L | 1/2 ) times to complete the proof.

Proof.[Proof of Proposition 2.5.1] We prove the first statement, the argument for the other ones being analogous. Fix σ ∈ R lc . Recall that we denote by T 1 the time of the first jump. By the strong Markov property at time T 1 and by Assertion 2.5.2,

where R β is a remainder whose absolute values is bounded by C 0 δ 1 (β), and σ -, resp. σ + , is the configuration obtained from σ by flipping to -1 the attached 0-spin, resp. by flipping to 0 the unique -1 spin with two 0-spins as neighbors.

The configuration σ -has n 0 (n 0 + 1) 0-spins which form a n 0 × (n 0 + 1)-rectangle in a background of -1. Hence, by Corollary 2.5.6,

On the other hand, by Corollary 2.5.11,

The first statement of the proposition follows from these estimates and from the fact that δ 2 (β) < δ 5 (β), because 4 -n 0 h > 2, and δ 3 (β) < C 0 δ 5 (β) for β large enough by (2.32).

Proof of Theorems 2.2.1 and 2.2.2

The proofs of Theorems 2.2.2 and 2.2.1 are based on Propositions 2.4.1 and 2.5.1. By Proposition 2.5.1, there exists a finite constant C 0 such that

. By Proposition 2.4.1, the previous expectation is equal to

where ε(β) has been introduced in (2.18).

By (2.35), this expression is equal to

where the value of R(β) has changed but not its bound. Therefore, 1 -

Since, by Proposition 2.4.1,

replacing on the right-hand side σ∈R l q(σ) by 1 -σ∈R s q(σ) -R (β), where the absolute value of R (β) is bounded by ε(β), we conclude that

as claimed.

The convergence of the trace process

In this section, we examine the evolution of the trace of σ t on M = {-1, 0, +1} under the hypotheses of Theorem 2.2.2. Denote by η t the trace of σ t on M. We refer to Section 2.3 for a precise definition. By [2, Proposition 6.1], η t is an M-valued, continuous-time Markov chain. Recall the definition of θ β given in (2.7). Proposition 2.7.1. As β ↑ ∞, the speeded-up Markov chain η(θ β t) converges to the continuous-time Markov chain on M in which +1 is an absorbing state, and whose jump rates r(η, ξ), are given by r(-1, 0) = r(0, +1) = 1 , r(-1, +1) = r(0, -1) = 0 .

The proof of this proposition, presented at the end of this section, relies on estimation of capacities. We start characterizing the distribution of σ(H B + ) when the process starts from -1. Recall the definition of δ 2 (β) introduced just before Lemma 2.5.5.

Lemma 2.7.2. There exists a finite constant

Proof. Fix a reference configuration σ * in R a . By (2.11) and by definition of the capacity,

By reversibility, the numerator of this expression is equal to

By Assertions 2.5.2 and 2.5.3, with a probability close to 1, either a negative spin next to the attached 0-spin flips to 0 or the attached 0-spin flips to -1. In the first case, as the process left the valley V -1 introduced at the beginning of Section 2.4, it will hit B + before reaching -1. In the second case, applying Lemma 2.5.5 repeatedly yields that the process reaches -1 before hitting B + . Hence, by these three results,

where

and since µ β (σ) = µ β (σ * ), we conclude that

where the absolute value of R β is bounded by

To complete the proof of the lemma, it remains to multiply both sides of the penultimate identity by |R a |.

It follows from the proof of the previous lemma and the identity

Proposition 2.7.3. For any configuration η ∈ R l and any configuration ξ ∈ R l 0 ,

Proof. We prove below the first idendity of the proposition, the one of the second being analogous. We first claim that cap(-1, {0, +1}) = cap(-1, B + )

(2.37) Indeed, since starting from -1 the process hits B + before {0, +1}, by the strong Markov property we have that

By (2.11), we may rewrite the previous expression as

This proves (2.37) in view of the definition (2.8) of the capacity. By (2.36) and (2.37), for any configuration

where |R

(1)

] for β large. Consider the sum. By Proposition 2.4.1, we may ignore the terms σ ∈ R a . On the other hand, Proposition 2.5.1 provides the asymptotic value of P σ [H {0,+1} < H -1 ] for σ ∈ R a . Putting together these two result yields that the sum is equal to

where the absolute value of the remainder R

(2)

. By Lemma 2.7.2, this expression is equal to 1 2

where |R

(3)

The first assertion of the proposition follows from the previous estimates.

The same proof yields that for any configuration

In particular,

Corollary 2.7.4. We have that

where θ β has been introduced in (2.7).

Proof. Fix η ∈ R a and ξ ∈ R a 0 . By definition of θ β , the expression appearing in the statement of the corollary can be written as

•

By the previous lemma, the product of the second and third expression converges to 1, while the first and fourth term cancel.

This yields (2.40) and completes the proof of the lemma.

The time spent out of M

We prove in this section that the total time spent out of M by the process σ(tθ β ) is negligible. Unless otherwise stated, we assume that the hypotheses of Theorem 2.2.2 are in force. The proof of this proposition is divided in several steps. Suppose that the process starts from -1. In this case, we divide the time interval [0, t] in 5 pieces, and we prove that the time spent in ∆ in each time-interval [0,

The proof of the last step requires the introduction of the valley of +1 which is slightly different from V -1 and V 0 . Denote by V +1 the valley of +1. This is the set constituted of all configurations which can be attained from +1 by flipping n 0 (n 0 +1) or less spins of +1. The boundary of this set, denoted by B + +1 , is formed by all configurations which differ from +1 at exactly n 0 (n 0 + 1) sites. The configuration with minimal energy in B + +1 is the one where n 0 (n 0 + 1) 0-spins form a n 0 × (n 0 + 1)rectangle. Denote the set of these configurations by R +1 . Fix η ∈ R +1 and ξ ∈ R a and note that H(η) -H(+1) > H(ξ) -H(-1) .

(2.41) Thus V +1 is a deeper valley than V -1 or V 0 . Indeed, according to [2, Theorem 2.6], the depth of the valley V +1 is given by µ β (+1)/cap(+1, B + +1 ). As in the proof of (2.36), or by applying the Dirichlet and the Thomson principles, we have that cap(+1, B + +1 ) is of the order of |Λ L | µ β (η) for η ∈ R +1 . Hence the depth of the valley V +1 is of the order of e [H(η)-H(+1)]β /|Λ L |. By (2.41), the definition (2.7) of θ β , and Proposition 2.7.3, this expression is much larger than θ β , which is of the same order of the depth of the valley V -1 .

In particular, for all t > 0 lim β→∞

.42)

We turn to the proof of Proposition 2.8.1. We first show that conditioned to the valley V -1 , the measure µ β is concentrated on the configuration -1. The same argument yields that this result holds for the pairs (0, V 0 ), (+1, V +1 ). Lemma 2.8.2. Suppose that (2.4) holds. Then, there exists a constant C 0 such that

for all β ≥ C 0 .

Proof. Fix 1 ≤ k ≤ N = n 0 (n 0 + 1) + 1, and denote by A k the configurations in V -1 with k spins different from -1. The ratio appearing in the statement of the assertion is equal to

In this equation, A 0 k represents the configurations in V -1 with k spins equal to 0, and we applied Assertion 2.4.5.

Proof. Consider a configuration σ ∈ R. Applying Lemma 2.5.5 repeatedly yields that P σ [H B + < H -1 ] → 0. We may therefore include the indicator of the set {H -1 < H B + } inside the expectation appearing in the statement of the lemma. After this inclusion, we may replace H -1 by H B + ∪{-1} . At this point, it remains to estimate (1/θ β ) E σ [H B + ∪{-1} ]. Since the process is stopped as it reaches B + , we may replace σ t by the reflected process at V -1 . After this replacement, bound H B + ∪{-1} by H -1 .

We need therefore to estimate (1/θ

. By [2, Proposition 6.10] and by definition of θ β , this expression is bounded by cap(σ * , -1) µ β (-1)

,

where σ * is a configuration in R a . By Lemma 2.8.2, µ V (-1) → 1, while by the proofs of Lemma 2.4.2 and 2.4.3,

Consider now a configuration σ ∈ R s 2 . Denote by A the [n 0 × (n 0 + 2)]-rectangle obtained from the set of 0-spins of σ by completing the line or the row where the two extra spins sit. Denote by ξ the configuration where each site in A has a 0-spin, all the other ones being -1.

As in the proof of Lemma 2.5.5, we define the valley V ξ of ξ in two stages. Fix 0 ≤ k ≤ n 0 . We first flip sequentially k spins of A from 0 to -1. At each step we only flip a 0-spin if it is surrounded by two -1-spins. The set of all configurations obtained by such a sequence of k flips is represented by G k . In particular, since at the beginning we may only flip the corners of A, G 1 is composed of the four configurations obtained by flipping to -1 one corner of A. Let G = ∪ 0≤k<n 0 G k , and note that G n 0 has not been included in the union.

The second stage in the construction of the valley V ξ consists in flipping a spin of a configuration in G. More precisely, denote by B all configurations which are not in G, but which can be obtained from a configuration in G by flipping one spin. The set B is interpreted as the boundary of the valley V ξ := G ∪ B and it contains G n 0 .

Note that σ belongs to G and that starting from σ the process hits B before -1, so that

Replacing σ t by the process reflected at V ξ , applying [2, Proposition 6.10], and estimating the capacities yield that the first term divided by θ β converges to 0 as β → ∞.

We turn to the second term. We may insert the indicator function of the set {H B = H {η (1) ,η (2) } }, where η (1) , η (2) are the configurations obtained from ξ by flipping to -1 a line or a row of length n 0 of the rectangle A. After this insertion, the strong Markov property yields that the second term of the previous displayed equation is bounded by

Since the configurations η (1) , η (2) belong to R, the result follows from the first part of the proof.

As µ V is the stationary state for the reflected process, this expression is equal to

By Lemma 2.8.2, for V +1 instead of V -1 , this expression vanishes as β → ∞.

Proof.[Proof of Proposition 2.8.1] Lemmata 2.8.5 and 2.8.6 show that the time spent on ∆ until the process reaches 0 is negligible. We may repeat the same argument to extend the result up to the time where the process reaches +1. Once the process reaches +1, we apply Lemma 2.8.7.

Proof of Theorem 2.2.3

Unless otherwise stated, we assume throughout this section that the hypotheses of Theorem 2. (2.44)

We present the proof for η = -1. The proof for η = 0 is identical. The one for η = +1 is even simpler because the valley is deeper.

Lemma 2.9.1. Under P -1 , the random variable 3 H B + /θ β converges to a mean one exponential random variable.

Proof. As we are interested in H B + , we may replace the process σ t by the reflected process at V -1 , denoted by η t . The proof is based on [2, Theorem 2.6] applied to the triple ({-1}, V -1 \ B + , -1). We claim that the process η t fulfills all the hypotheses of this theorem. Condition (2.14) is satisfied because the set {-1} is a singleton, and condition (2.15) is in force in view of Lemma 2.8.2. Therefore, by this theorem, the triple ({-1}, V -1 \ B + , -1) is a valley of depth µ V (-1) cap V (- By the previous lemma, for 2δ ≤ s ≤ 3δ, P -1 σ(sθ β ) ∈ ∆ = P -1 σ(sθ β ) ∈ ∆ , H B + ≥ 4δθ β + R β,δ , for some remainder R β,δ which vanishes as β → ∞ and then δ → 0. On the set {H B + ≥ 4δ}, we may replace the process σ t by the reflected process η t and bound the first term by

because µ V is the stationary state. By Lemma 2.8.2 this expression vanishes as β → ∞, which proves (2.44) for η = -1.