Quentin Berger

Maha Khatib

Giambattista Giacomin

DNA MELTING STRUCTURES IN THE GENERALIZED POLAND-SCHERAGA MODEL

Keywords: 2010 Mathematics Subject Classification: 60K35, 82D60, 92C05, 60K05, 60F10 DNA Denaturation, Polymer Pinning Model, Two-dimensional Renewal Processes, Sharp Deviation Estimates, non-Cramér regime, Condensation phenomena

Introduction

The Poland-Scheraga (PS) model [START_REF] Poland | Theory of helix-coil transitions in biopolymers;: Statistical mechanical theory of order-disorder transitions in biological macromolecules[END_REF][START_REF] Blake | Thermal stability of DNA[END_REF][START_REF] Einert | Secondary structure formation of homopolymeric singlestranded nucleic acids including force and loop entropy: implications for DNA hybridization[END_REF] played and still plays a central role in the analysis of DNA denaturation (or melting): double stranded DNA melts into two single stranded DNA polymer chains at high temperature. The success of the model is partly due to the fact that it is exactly solvable when the heterogeneous character of the DNA is neglected. Moreover, solvability has an interest on its own, from a more theoretical standpoint: phase transition and critical phenomena in the PS model are completely understood [START_REF] Fisher | Walks, walls, wetting, and melting[END_REF][START_REF] Giacomin | Random polymer models[END_REF]. However, the PS model is an oversimplification in many respects: it deals with two strands of equal length and it does not allow slippages of the two chains. These simplifications make the model one dimensional, and solvability becomes less surprising. What is instead surprising is that a natural generalization [START_REF] Garel | On the role of mismatches in DNA denaturation[END_REF][START_REF] Garel | Generalized Poland-Scheraga model for DNA hybridization[END_REF][START_REF] Neher | Intermediate phase in DNA melting[END_REF] -called generalized Poland-Scheraga (gPS) model -fully overcomes these limitations, retaining the solvable character in spite of the substantially richer variety of structures that it displays. In [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF] a mathematical approach to the gPS model is developed and it is pointed out that it can be represented in terms of a two dimensional renewal process, much like the PS model can be represented in terms of a one dimensional renewal. The solvable character of both models is then directly related to their renewal structure. The growth in complexity from PS to gPS models is nevertheless considerable: the key feature of PS and gPS is the presence of a localization transition, corresponding to the passage from separated to bound strands, and for the gPS there are three, not only one, types of localized trajectories (or configurations). This has been first pointed out, at least in part, in [START_REF] Neher | Intermediate phase in DNA melting[END_REF], where one can find theoretical arguments (based also on a Bose-Einstein condensation analogy) and numerical evidence that "suggest that a temperature-driven conformational transition occurs before the melting transition" [17, p.3].

In this work we fully characterize the possible localized configurations. The transitions between different types of configurations have been already studied at the level of the free energy in [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF] where these phenomena have been mathematically identified and interpreted in a Large Deviations framework in terms of Cramér and non-Cramér strategies. This will be explained in detail below. Here we content ourselves with pointing out that a full analysis of the Cramér regimes is given in [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF]. However, the non-Cramér regime, where the condensation phenomena happen, requires a a substantially finer analysis -moderate deviations and local limit estimates -at the level of the bivariate renewals. These estimates, to which much attention has been devoted in the literature in the one dimensional set-up (see [START_REF] Armendáriz | Conditional distribution of heavy tailed random variables on large deviations of their sum[END_REF][START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF] and references therein), are lacking to the best of our knowledge for higher dimensional renewals and they are not straightforward generalizations. They represent the technical core of this paper.

1.1. The Model and some basic results. We introduce the model in detail only from the renewal representation. The link with the original representation of the model is summed up in Fig. 1 and its caption, and we refer to [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF] for more details.

We consider a persistent bivariate renewal process τ = {(τ

n , τ

n )} n≥0 , that is a sequence of random variables such that τ 0 = (0, 0), {τ nτ n-1 } n=0,2,... is IID and such that the inter-arrival law -i.e. the law of τ 1 -, takes values in

N 2 := {1, 2, • • • } 2 .
We set P(τ 1 = (n, m)) = K(n + m) with

K(n) := L(n) n 2+α , (1.1)
for some α > 0 and some slowly varying function L(•). Moreover n,m K(n + m) = 1 since we assumed the process to be persistent. We consider two versions of the model: constrained and free. The partition function of the constrained model, or constrained partition function, can be written as

Z c N,M,h := N ∧M n=1 l∈N n |l|=N t∈N n |t|=M n i=1 exp(h)K(l i + t i ) , (1.2)
where h ∈ R is the binding energy, or pinning parameter. The partition function of the free model, or free partition function, is defined by

Z f N,M,h := N i=0 M j=0 K f (i)K f (j)Z c N -i,M -j,h , (1.3)
where K f (n) := L(n)n -α for some α ∈ R and slowly varying function L(•). We assume that K f (0) = 1 just to prevent this constant from popping up in various formulas: this choice has the side effect of making clear that K f (•) is not a probability.

In [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF] it is shown that for every h and every γ > 0 In particular we see that (1, 1) renewal increments (or inter-arrivals) correspond to bound base pairs and all other increments (i, j) correspond to unbound regions in the bulk, that we call loops (of total length i + j, with length i in the first strand and j in the second strand). The term unbound is rather reserved to the terminal portion of the polymer: we refer to the free ends as unbound strands. Throughout this work, a polymer trajectory is always given in the renewal representation: it is therefore just a point process in the plane.

f γ (h) := lim N,M →∞ M/N →γ 1 N log Z c N,M,h = lim N,M →∞ M/N →γ 1 N log Z f N,M,h < ∞ , (1.4)
which says that the free energy (density) of free and constrained models, with binding energy h and strand length asymptotic ratio equal to γ, coincide. A number of basic properties of h → f γ (h) are easily established, notably that it is a convex non decreasing function, equal to zero for h ≤ 0 and positive for h > 0. This already establishes that h = 0 is a critical point, in the sense that f γ (•) is not analytic at the origin. But [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF] is not limited to results on the free energy: associated to Z c N,M,h and Z f N,M,h there are two probability measures, that we denote respectively by P c N,M,h and P f N,M,h . They are point measures, like the renewal processes on which they are built. It is standard to see that ∂ h f γ (h) (which exists except possibly for countably many values of h) yields the N → ∞ limit of the expected density of points (under P c N,M,h or P f N,M,h ). Hence for h < 0 the density is zero, while for h > 0 the density is positive. This tells us that we are stepping from a regime in which the two strands are essentially fully unbound to a regime in which they are tightly bound. In [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF] results go well beyond this: it is in particular proven that for h < 0 the number of renewal points is O(1) and these points are all close to (0, 0) or (N, M ) (see Fig. 2). In the polymer representation, this means that the two DNA strands are completely unbound, except for a few contacts between the bases just close to the extremities. More precisely, it was found in [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF] that in the free case, if α < 1 + α/2 the two strands are free except for O(1) contacts close to the origin, and if α > 1 + α/2 the two free ends are of length O(1) and a large loop appears in the system, see A schematic image of the two types of observable trajectories of the free gPS model in the delocalized (denaturated, melted) regime, according to whether the exponent α is smaller (left picture) or larger (right picture) than 1 + α/2. In the constrained case only the trajectory on the right is observed, and the small free tails are reduced to zero. This case is treated in [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF].

On the other hand for h > 0 the situation is radically different. This has been analyzed in [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF] but only in the Cramér regime. We are now going to discuss this in details.

Binding strategies.

A way to get a grip on what is going on for h > 0 is to observe that we can make the elementary manipulation: for every non negative λ 1 and λ 2

Z c N,M,h = e N λ 1 +M λ 2 N ∧M n=1 l∈N n |l|=N t∈N n |t|=M N i=1 exp (h -λ 1 l i -λ 2 t i ) K(l i + t i ) . (1.5)
Since h > 0 we identify a family, in fact a curve in [0, ∞) 2 , of values of (λ 1 , λ 2 ) such that

∞ l,t=1 exp (h -λ 1 l -λ 2 t) K(l + t) = 1 , (1.6)
and (1.6) clearly defines a probability distribution that is an inter-arrival distribution for a new bivariate renewal process. At this point is not too difficult to get convinced that Z c N,M,h is equal to e N λ 1 +M λ 2 times the probability that this new renewal hits (N, M ) (we call this probability target probability). If we are able to choose (λ 1 , λ 2 ) so that the logarithm of the target probability is o(N ), then of course f γ (h) = λ 1 + γλ 2 . This can actually be done: it amounts to solving a variational problem and the uniqueness of the optimal (λ 1 , λ 2 ) follows by convexity arguments. However the solution may be qualitatively different for different values of h:

(1) the optimal (λ 1 , λ 2 ) belong to (0, ∞) 2 , so both components of the inter-arrival law of the arising renewal have distributions that decay exponentially. We call this Cramér regime because the tilt of the measure (in both components) is efficient in targeting the point (N, M ) to which we are aiming at; (2) either λ 1 or λ 2 is zero, so only one component of the arising inter-arrival law is exponentially tight. For the sake of conciseness we call this for now non-Cramér regime because the tilt of the measure (in only one of the component) is only partially successful in making the renewal τ reach and visit the point (N, M ). To be precise there is in reality a boundary region between the two regimes, and the notion of non-Cramér regime will be made more precise just below -this regime is the main issue of this work -so we will not dwell further on this right now.

A full treatment of the Cramér regime is given in [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF], and the results can be resumed as follows: all loops are small, in fact the largest is O(log N ), and the unbound strands (the free ends) are of length O(1) -see the leftmost case in Fig 3 . In the present work, we focus on the non-Cramér regime and the reader who wants to have an anticipation on the results can have a look Fig 3 .   1.3. The non-Cramér regime. In order to make as explicit as possible for which values of h > 0 the system is in the non-Cramér regime, let us define n(h) > 0 as the unique solution of

∞ n,m=1 K(n + m) exp (-nn(h)) = exp(-h) .
(1.7)

This computation amounts to solving the variational problem we were after, in the case in which the problem is not solvable in (0, ∞) 2 and the optimal tilt of the measure involves only one of the two components. From (1.7) one can extract a number of properties of n(•): it is a real analytic, positive, convex and increasing function [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF]. We insist on the fact that, in spite of being defined for every h > 0, n(h) is not always equal to the free energy. More precisely in [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF] it is shown that

f γ (h) = n(h) if and only if γ / ∈ (1/γ c (h), γ c (h))
, where

γ c (h) := n,m mK(n + m) exp(-nn(h))
n,m nK(n + m) exp(-nn(h))

.

(1.8)

We refer to [START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF] for more details on the form of the function γ c (•) and the switching phenomena between the Cramér and the non-Cramér regime. In this work, and without loss of generality (by symmetry), we will consider only the case γ > γ c (h). To be precise we will rather consider the case γ ≥ γ c (h) because the phenomenology observed for γ > γ c (h), that is for Mγ c (h)N ≥ cN for some c > 0 persists also in a part of the window Mγ c (h)N = o(N ) and we will analyze the model also in this window. In different terms: the analysis in the Cramér regime is a Large Deviations analysis, but the whole non-Cramér regime is equivalent from the Large Deviations viewpoint (the issues there are about sharp deviations). So there isn't much conceptual difference between Mγ c (h)N ≥ cN and Mγ c (h)N = o(N ), up to when Mγ c (h)N grows too slowly, as we shall see.

Crucial for us is the probability distribution Kh (•, •) defined by

Kh (n, m) = K(n + m)e h-nn(h) , (1.9)
which, as announced informally just above, allows to write the partition function as

Z c N,M,h = exp(N n(h))P ((N, M ) ∈ τh ) , (1.10)
where τh is the bivariate renewal process with inter-arrival distribution Kh (•, •), i.e.

P (τ h ) 1 = (n, m) = Kh (n, m) = L(n + m) (n + m) 1+α e h-nn(h) . (1.11)
Next, we are going to have a closer look at this renewal process.

1.4. On the bivariate renewal τh . Let us write for conciseness n(h) = n h (a practice that we will pick up again in the proofs), and drop the dependence on h in τh : τ = (τ (1) , τ (2) ). In view of (1.11), it is clear that the distribution of this process is not symmetric, we have the marginals

P τ (1) 1 = n = m≥1 K(n + m) exp(h -n h n) n→∞ ∼ exp(h) (1 + α) L(n) n 1+α e -n h n , P τ (2) 1 = m = n≥1 K(n + m) exp(h -n h n) m→∞ ∼ exp(h) exp(n h ) -1 L(m) m 2+α .
(1.12)

Let us also denote (the dependence in h is implicit)

μ1 := E[τ (1) 1 ] < +∞, μ2 := E[τ (2) 1 ] < +∞ , (1.13)
so that γ c (h) = μ2 /μ 1 , cf. (1.8).

We notice that the process τ (1) has moments of all orders, and so {τ

n } n=0,1,... is in the domain of attraction of a normal law: we denote a [START_REF] Armendáriz | Conditional distribution of heavy tailed random variables on large deviations of their sum[END_REF] n := √ n the scaling sequence for τ (1) n . On the other hand, the process {τ

(2) n } n=0,1,... is in the domain of attraction of an α 2 -stable law, with α 2 := (1 + α) ∧ 2 > 1: its scaling sequence a (2) n verifies L(a (2) n )(a (2) n ) -α 2 ∼ 1 n if α 2 < 2 and σ(a (2) n )(a (2) n ) -2 ∼ 1 n if α 2 = 2 (1.14) where σ(n) := E τ (2) 1 2 1 {τ (2) 1 n} , (1.15)
and diverges as a slowly varying function if E[(τ

(2)

1 ) 2 ] = +∞ (with σ(n)/L(n) → +∞ [3]). Note that σ(n) converges if α > 1. In particular, a (2)
n is regularly varying with exponent 1/α 2 = (1 + α) -1 ∨ (1/2).

As an additional relevant definition, we select a sequence {m

n } n=1,2,... satisfying

P τ (2) 1 > m (2) n n→∞ ∼ 1 n , (1.16)
so that m

n gives the order of max 1 j n {τ

(2)

j - τ (2) j-1 }. We stress that m (2)
n is regularly varying with exponent (1 + α) -1 , and that m

(2) n /a (2) n ∈ [1/c, c] for some c ≥ 1 if α < 1, but m (2) n /a (2) n → 0 when α 1: in any case, there is a constant c > 0 such that m (2) n ca (2) n for every n, i.e. m (2) n = O(a (2) n ).
1.5. Non-Cramér regime and big-jump domain. We drop the dependence of γ c (h) on h, and we set t N := Mγ c N .

(1.17) Of course, having γ > γ c means that t N /N ≥ c for some c > 0. But it is natural and essentially not harder to tackle the problem assuming only

t N /a (2) N → +∞ as N → ∞, (1.18)
with additionally, in the case α 1 (recall the definition of σ(n) after (1.14))

t N a (2) N 2 σ(a N ) σ(t N ) N →∞ ∼ t 2 N N σ(t N ) C 0 log N for a suitable choice of C 0 > 0 . (1.19) If α > 1, as well as if α = 1 and σ(n) = O(1) (i.e. if E[(τ (2) 1 ) 2 ] < ∞), (1.19) simply means that t N C 0 √ N log N with C 0 easily related to C 0 . Note also that (1.18) implies t N √ N log N if E[(τ (2) 1
) 2 ] = ∞. We stress that the constants C 0 depends only on K(•) and, for the interested reader, it can be made explicit by tracking the constants in (3.34) and (??) where the value of C 0 is used. This assumption is made to be sure that we lie in the so called big-jump domain, as studied for example in the one-dimensional setting in [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF]: in our model it simply means that deviations -and the event we focus on is (N, M ) ∈ τ -are realized by an atypical deviation on just one of the increment variables τi+1τi . As we shall, this happens just under the assumption (1.18) for α < 1 and this condition is optimal (see Appendix B.1). For the case α ≥ 1 the extra condition (1.19) is not far from being optimal, but it is not: we discuss this point in Appendix B.2, but we do not treat it in full generality because it is a technically demanding issue that leads far from our main purposes.

1.6. Mais results I: polymer trajectories. Let us now introduce two fundamental events in an informal, albeit precise, fashion. The two events will be rephrased in a more formal way in (2.9), once further notations will have been introduced. Choose sequences of positive numbers

{u N } N =1,2,... , {m + N } N =1,2,... , {a + N } N =1,2,... and { a + N } N =1,2,... such that u N 1 , t N m + N m (2)
N , t N a + N a (2) N and t N a + N a (2)
N .

(1.20)

In practice, and to optimize the result that follows,

u N , m + N /m (2)
N , a + N /a (2)
N and a + N /a

(2) N should be chosen tending to ∞ in an arbitrarily slow fashion.

We then define the Big Loop event E

(N )

BL to be the set of trajectories such that (1) there is one loop of size larger than t Na + N and smaller than t N + a + N , so that, to leading order, it is of size t N ;

(2) all other loops are smaller than m + N (hence there is only one largest loop); (3) the length of neither of the two unbound strands is larger than u N .

The (large or macroscopic) Unbound Strand event E (N )

US is instead the set of trajectories such that

(1) all loops are smaller than m + N ; (2) the length of the unbound portion of the shorter strand does not exceed u N ;

(3) the length of the unbound portion of the longer strand is larger than t Na + N and smaller than t N + a + N , so that, to leading order, it is of size t N . Note that

E (N ) BL ∩ E (N )
US = ∅ except, possibly, for finitely many N : the two conditions (1) are incompatible. We refer to Fig. 3 for a schematic image of these two events. Theorem 1.1. Under assumptions (1.18), (1.19) and (1.20) we have that lim

N →∞ P f N,M,h E (N ) BL ∪ E (N ) US = 1 . (1.21) (1) Moreover, if α < 1 (and hence j K f (j) = ∞) then lim N →∞ P f N,M,h E (N ) US = 1 . (1.22) (2) If j K f (j) < ∞ (and hence α ≥ 1) then P f N,M,h E (N ) US N →∞ ∼ 1 1 + Q N and P f N,M,h E (N ) BL N →∞ ∼ Q N 1 + Q N , (1.23)
with

Q N := c h N (t N ) -(2+α)+α L(t N ) L(t N ) and c h := e h ∞ j=0 K f (j) μ1 (e n(h) -1)
.

(1.24)

For conciseness the case α = 1 with j K f (j) = +∞ is not included in Theorem 1.1, but its is treated in full in Appendix A. It is a marginal case in which an anomalous behavior appears: a big loop and a large unbound strand may coexist. ). In the constrained case the Unbound Strand event is not observed, and the free tails are of course absent. What cannot be appreciated in this schematic view is the fact that the small loop distribution has exponential tail in the Cramér regime (hence the largest is O(log N )) and that it has power law tail in the non-Cramér regime (hence the largest is O(N a ) for some a ∈ (0, 1): O(mN ) to be precise).

It is worth pointing out that, in most of the cases, the expressions in (1.23) have a limit -at least if {t N } N =1,2,... is not too wild (regularly varying is largely sufficient) -and it is either one or zero. In particular when t N ∼ cN for some c > 0 we have lim

N →∞ P f N,M,h E (N ) US =    1 if α < α + 1 or if α = α + 1 and L(t) t→∞ L(t), 0 if α > α + 1 or if α = α + 1 and L(t) t→∞ L(t). (1.25)
(This is true also in the case α = 1 with j K f (j) = +∞, see (A.6)-(A.7)). Note that in the case in which α = α + 1 and the ratio of the two slowly varying function has a limit which is neither 0 nor ∞, the limit of the probability of the unbound strand event exists and it is an explicit value in (0, 1). Let us also give the following complement to Theorem 1.1 for the constrained case.

N →∞ P c N,M,h E (N ) BL,0 = 1 , (1.26)
where

E (N ) BL,0 is the event E (N )
BL with the more stringent condition that the the length of both the unbound strands is 0.

Main results II: sharp estimates on the partition functions

In this section, we give the asymptotic behavior of e -N n h Z c N,M,h = P((N, M ) ∈ τ ) in the big-jump domain. Then we present the asymptotic behavior of Z f N,M,h . Both in the constrained and free case we also give more technical estimates that identify some events to which we can restrict the partition functions without modifying them in a relevant way. Theorem 1.1 turns out to be a corollary of these technical estimates, as we explain in the final part of the section.

In this section and in the rest of the paper we deal with order statistics and we introduce here the relative definitions. Consider the (non-increasing) order statistics

M 1,k , M 2,k , . . . , M k,k of the IID family {τ (2) j - τ (2) j-1 } j=1,.
..,k . In particular M 1,k is a maximum of this finite sequence. We will consider the order statistics also for k random, notably for k = κ N , with

κ N := max{i : τ (1) i ∈ [0, N ]}. (2.1)
2.1. On the constrained partition function. We start with an important estimate for the constrained partition function (more precisely for P((N, M ) ∈ τ ), the the renewal mass function), that is essential for the study of the free partition function, as one can imagine from its definition (1.3). It is worth insisting on the link between P and the measure we are interested in for the constrained case:

P c N,M,h (•) = P( • |(N, M ) ∈ τ ) . (2.2)
Theorem 2.1. Assume that α > 0 and that (1.18) holds. Moreover if α 1 assume also (1.19). Recall that τ has inter-arrival distribution (1.11). Then (recall that M = γ c N +t N ) we have that

P (N, M ) ∈ τ = P τ (1) κ N = N, τ (2) κ N = M N →∞ ∼ N μ2 1 P τ (2) 1 = t N . (2.3)
Moreover, for every η ∈ (0, 1) there exist ε 0 > 0 such that for every ε ∈ (0, ε 0 ) and N sufficiently large (how large may depend on ε), we have

P M 1,κ N -t N ∈ - a (2) N ε , a (2)
N ε , M 2,κ N ≤ m (2) N ε , τ (1) κ N = N, τ (2) κ N = M ≥ ≥ (1 -η) N μ2 1 P τ (2) 1 = t N . (2.4)
Let us stress that the bivariate renewal process τh falls in the domain of attraction of an (α 1 = 2, α 2 ) stable distribution (see e.g. [START_REF] Resnick | A bivariate stable characterization and domains of attraction[END_REF] or [START_REF] De Haan | Domains of attraction and regular variation in R d[END_REF]). We have, as n → ∞,

that τ (1) n -μ 1 n a (1) n , τ (2) n -μ 2 n a (2) n n=1,2,... converges in distribution to a non-degenerate (2, α 2 )- bivariate stable law.
We mention that a bivariate local limit theorem is given in [START_REF] Doney | A bivariate Local Limit Theorem[END_REF] and multivariate (ddimensional) renewals are further studied in [START_REF] Berger | Strong renewal theorems and local large deviations for multivariate random walks and renewals[END_REF]: local large deviation estimates are given, as well as strong renewal theorems, i.e. asymptotics of P((n, m) ∈ τ ) as (n, m) → ∞, when (n, m) is close to the favorite direction -i.e. the line t → tE[τ 1 ], close to meaning at distance of the order of the fluctuations around that direction. We refer to [START_REF] Berger | Strong renewal theorems and local large deviations for multivariate random walks and renewals[END_REF] for further details (rough estimates when (n, m) is away from the favorite direction are also given). We stress that the multivariate case we consider is very special, and that here we need and obtain estimates that are (much) sharper than what one finds in the literature we have cited.

2.2. On the free partition function. We now give the behavior of the free partition function and identify trajectories contributing the most to it. Let us introduce some notations:

V (N ) 1 := N -τ (1) κ N , V (N ) 2 := M -τ (2) κ N , (2.5)
the lengths of the free parts, see Fig 1 .  For a set A of allowed trajectories, we define Z f N,M,h (A) the partition function restricted to trajectories in A (by restricting the summation over subsets of {1, . . . N } × {1, . . . M } to those in A). For example,

Z f N,M,h ((N, M ) ∈ τ ) = Z c N,M,h . We set K := ∞ j=1 K f (j) if the sum is finite, and we set K = 0 if j K f (j) = +∞.
Theorem 2.2. Assume that α > 0 and (1.18). Moreover if α 1 assume also (1.19)

and if α = 1 assume that j K f (j) < ∞. Then for N → ∞ e -N n(h) Z f N,M,h = (1 + o(1))K N μ2 1 i≥0 K f (i)e -in(h) P τ (2) 1 = t N + (1 + o(1)) 1 μ1 i 0 K f (i)e -in h K f (t N ) . (2.6)
Moreover, for every η ∈ (0, 1) there exist ε 0 > 0 such that for every ε ∈ (0, ε 0 ) and N sufficiently large (how large may depend on ε), such that

e -N n h Z f N,M,h V (N ) 1 , V (N ) 2 1 ε , M 1,κ N ∈ t N - a (2) N ε , t N + a (2) N ε , M 2,κ N ≤ m (2) N ε (1 -η)K N μ2 1 i 0 K f (i)e -in h P τ (2) 1 = t N ; (2.7) e -N n h Z f N,M,h V (N ) 1 1 ε , V (N ) 2 ∈ t N - a (2) N ε , t N + a (2) N ε , M 1,κ N m (2) N ε (1 -η) 1 μ1 i 0 K f (i)e -in h K f (t N ) . (2.8)
We remark that when α < 1 (so K = 0), the right-hand side of (2.6) reduces to one term and (2.7) becomes trivial. The case α = 1 with j K f (j) = +∞ is treated in Theorem A.1.

2.3.

Back to the Big Loop and Unbound Strand events. The notations we have introduced allow a compact formulation of the two key events of Theorem 1.1:

E (N ) BL = M 1,κ N ∈ [t N -a + N , t N + a + N ], M 2,κ N < m + N , max V (N ) 1 , V (N ) 2 ≤ u N , E (N ) US = M 1,κ N < m + N , V (N ) 1 ≤ u N , V (N ) 2 ∈ [t N -a + N , t N + a + N ] .
(2.9)

Proof of Theorems 1.1 and 1.2. This is just a book-keeping exercise using the three estimates in Theorem 2.2 (or Theorem 2.1 for Theorem 1.2), together with the definition of K f (t N ) and the estimate of P(τ

1 = t N ) in (1.12).

2.4.

A word about the arguments of proof and organization of the remaining sections. As we pointed out at the beginning of the introduction, condensation phenomena are widely studied in the mathematical literature (see [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF][START_REF] Armendáriz | Conditional distribution of heavy tailed random variables on large deviations of their sum[END_REF] and references therein), but not in the multivariate context. The full multivariate context is the object of [START_REF] Berger | Strong renewal theorems and local large deviations for multivariate random walks and renewals[END_REF], where renewal estimates, i.e. estimates on P((n 1 , . . . , n d ) ∈ τ ), are given: the big-jump domain in treated in much greater generality, but not with the precision we require. Here we deal only with the very special bivariate case motivated by the application -which has the simplifying feature of being highly asymmetric so in the end we can reduce to a one dimensional renewal -and we need and find the exact local asymptotic behavior of the renewal function. One of the main difficulties we face is that, on the event (N, M ) ∈ τ , the number κ N of renewal points is random and highly constrained by this event. We show that in the big-jump domain considered in Section 1.5, the main contribution to the probability P((N, M ) ∈ τ ) comes from trajectories with a number of renewal points that is approximately

κ N = N/μ 1 + O( √ N ). For this number κ N , τ (1)
κ N does not have to deviate from its typical behavior to be equal to N , but τ (2) κ N has to deviate from its typical behavior to reach M and it does so by making one single big jump, of order t N + O(a

(2) N ). In this sense, if we accept that κ N is forced to be N/μ 1 + O( √ N ) by the condition N ∈ τ (1)
, we can focus on M ∈ τ (2) and the problem becomes almost one dimensional. This turns out to be a lower bound strategy: for a corresponding upper bound we have to show that all other trajectories bring a negligible contribution to P((N, M ) ∈ τ ).

In the rest of the paper, we estimate separately the constrained and free partition functions. We deal with the constrained partition function in Section 3: the main term (2.4) in Section 3.1 and the remaining negligible contributions in Section 3.2. The free partition function is dealt with in Section 4: the main terms (2.7) and (2.8) in Section 4.1 and the remaining negligible contributions in Section 4.2. In Appendix A we complete the analysis of the case α = 1. In Appendix B we discuss the transition from the big-jump regime (a single big jump, with a big deviation of just one of the two components) to the Cramér deviation strategy (no big jump).

To keep things simpler in the rest of the paper, and with some abuse of notation, we will systematically omit the integer part in the formulas.

3. The constrained partition function: proof of Theorem 2.1 3.1. Proof of the lower bound (2.4). We start by decomposing the event of interest according to κ N = k. The probability of such an event, restricted to {κ N = k}, becomes (recall that M = γ c N + t N )

P |M 1,k -t N | ≤ a (2) N ε , M 2,k ≤ m (2) N ε , τ (1) k = N, τ (2) k = M = P k j=1 τ (2) j - τ (2) j-1 ∈ t N + I N , max i =j τ (2) i - τ (2) i-1 ≤ m (2) N ε , τ (1) k = N, τ (2) k = M , (3.1)
where we defined

I N := {-a (2)
N /ε, . . . , a

N /ε}. Since t Na

N /ε is larger than m

(2)

N /ε (recall that m (2) n ca (2)
n for every n, and (1.18)) for N sufficiently large, the union in the right-hand side of (3.1) is a union of disjoint events that have all the same probability. This term is equal to

k P τ (2) 1 ∈ t N + I N , max i=2,...,k τ (2) i - τ (2) i-1 ≤ m (2) N ε , τ (1) k = N, τ (2) k = γ c N + t N = k y∈I N x∈N P τ (2) 1 = t N +y, τ (1) 1 = x P M 1,k-1 m (2) N ε , τ (1) k-1 = N -x, τ (2) k-1 = γ c N -y , (3.2)
where we have used that {(τ

(1) j - τ (1) 1 , τ (2) j - τ (2) 1 )} j=2,...,k and {(τ (1)
j , τ (2) j )} j=1,...,k-1 have the same law.

Since we are after a lower bound we may and do restrict the sum over x between 1 and 1/ε and y ∈ I N := {-a

(2) N /ε, . . . , a (2)
N /ε}. And using that P(τ (2) 1 = n) is regularly varying, we have that uniformly for such x and y ∈ I

N P τ (2) 1 = t N + y, τ (1) 1 = x ≥ (1 -δ N ) P τ (2) 1 = t N P τ (1) 1 = x τ (2) 1 = t N + y , (3.3)
where δ N = δ N (ε) ≥ 0 is such that lim N →∞ δ N = 0. If now we set p h (x) := (e n h -1)e -xn h , by using (1.12) we have that

P τ (1) 1 = x τ (2) 1 =t N + y = P τ (1) 1 = x, τ (2) 1
= t N + y P τ (2) 1 = t N + y ≥ (1 -δ N ) p h (x)K (t N + x + y) K (t N + y) ≥ (1 -δ N ) 2 p h (x) , (3.4)
possibly for a different choice of δ N = δ N (ε). Therefore, going back to (3.1) we see that (again, by redefining δ N )

P M 1,k ∈ t N + I N , M 2,k ≤ m (2) N ε , τ (1) k = N, τ (2) k = M ≥ (1 -δ N )k P τ (2) 1 = t N 1/ε x=1 p h (x)P M 1,k-1 m (2) N ε , τ (1) k-1 = N -x, τ (2) k-1 -γ c N ∈ I N . (3.5)
We now sum over the values of k and we restrict to k ∈ [(N/μ 1 )-N/ε, (N/μ 1 )+ N/ε]∩ Z := J N . Hence, redefining δ N , the left-hand side of (2.4) is bounded from below by

(1 -δ N )P τ (2) 1 = t N N μ1 1/ε x=1 p h (x)P ε (x) , (3.6)
where we defined, with n + N := max J N ,

P ε (x) := k∈J N P max i=1,...,n + N τ (2) i - τ (2) i-1 m (2) N ε , τ (1) k-1 = N -x, τ (2) k-1 -γ c N ∈ I N (3.7)
For P ε (x), we observe right away that by introducing also n - N := min J N -note that n ± N are equal to (N/μ 1 ) ± N/ε -we have

P ε (x) ≥ k∈J N P M 1,n + N m (2) N ε , τ (1) k-1 = N -x - k∈J N P M 1,n + N m (2) N ε , τ (1) k-1 = N -x, τ (2)
n + N -γ c N > a (2) N ε - k∈J N P M 1,n + N m (2) N ε , τ (1) k-1 = N -x, τ (2)
n - N -γ c N < a (2) N ε P (E 1 ∩ E 2 (x)) -P E + 3 -P E - 3 , (3.8)
where

E 1 := M 1,n + N m (2) N ε , E 2 (x) := ∃k ∈ J N such that τ (1) k-1 = N -x , (3.9
) and

E + 3 := τ (2) n + N -γ c N > a (2) N ε , E - 3 := τ (2) n - N -γ c N < a (2) N ε . (3.10)
We now estimate separately the probability of these events.

E 1 has probability close to one. For this, we use that P(τ

1 > n) is regularly varying with index (1 + α) -1 together with the definition (1.16) of m [START_REF] Berger | Strong renewal theorems and local large deviations for multivariate random walks and renewals[END_REF] n to obtain that for N larger than some constant N 0 = N 0 (ε) we have P τ

(2) 1 > 1 ε m (2) N 2ε 1+α N -1
. Therefore, we have for N N 0

P (E 1 ) 1 -2ε 1+α N -1 n + N e -3ε 1+α . (3.11)
where we used that n + N N and ε small. E 2 (x) has probability close to 1/μ 1 . The probability of E 2 (x) is estimated by writing

P (E 2 (x)) = P(N -x ∈ τ (1) ) - k<n - N P τ (1) k = N -x - k>n + N P τ (1) k = N -x P(N -x ∈ τ (1) ) -P τ (1) n - N N -1 ε -P τ (1) n + N N , (3.12)
where for the second term we used that

P ∃ k < n - N s.t. τ (1) k = N -x P τ (1) n - N N -x
together with the fact that x 1/ε (and similarly for the last term). First, because the inter-arrivals of τ (1) are exponentially integrable, |P(N ∈ τ (1) ) -1/μ 1 | ≤ exp(-cN ) for N ≥ N 0 with c > 0 and N 0 that depend on the inter-arrival law [START_REF] Kendall | Unitary dilations of Markov transition operators and the corresponding integral representation of transition probability matrices[END_REF]. Therefore, uniformly in x = 1, . . . , 1/ε, we have that for N sufficiently large

P(N -x ∈ τ (1) ) 1/μ 1 -e -cN/2 .
For the remaining terms in (3.12) it is just a matter of using the Central Limit Theorem. In fact, recalling that n - N = N/μ 1 -N/ε, we have

P τ (1) n - N N -1 ε = P τ (1) n - N -μ1 n - N μ1 ε -1/2 √ N -1 ε e -cε -1 , (3.13)
for N larger than some N 0 = N 0 (ε). On the other hand, we also have that P τ (1)

n + N N = P τ (1) n + N -μ1 n + N -μ1 ε -1/2 √ N e -c ε -1 , (3.14)
provided again that N is large enough. Therefore we have proven that for every η ∈ (0, 1) there exists ε 0 and N 0 : (0, 1) → N such that for ε ∈ (0, ε 0 ) and N ≥ N 0 (ε) we have min x=1,...,1/ε

P (E 2 (x)) ≥ 1 -η μ1 .
(3.15)

E ± 3 have a small probability. This is a consequence of the convergence to stable limit law. In fact, using that γ c = μ2 /μ 1 so that γ c N = μ2 n + N -μ2 N/ε, we get

E + 3 = τ (2) n + N -μ2 n + N > a (2) N ε -μ2 N/ε ⊂ τ (2) n + N -μ2 n + N > a (2) N 2ε , (3.16)
where the last inclusion holds provided that ε is sufficiently small, since there is a constant c such that a

(2) N c √ N for all N (we actually simply need N to be large if a

(2)

N / √ N → +∞ for N → ∞, which is the case when E[(τ (2)
1 ) 2 ] = +∞). Very much in the same way we get also to

E - 3 ⊂ τ (2) n - N -μ2 n - N < - a (2) N 2ε . (3.17)
Since (τ

(2) n ± N -μ2 n ± N )/a (2) n ± N
converges in law for N → ∞ to a stable limit variable Y , and using that a

(2)

N /a (2) n ± N → μ1/α 2 1 (since n ± N ∼ N/µ 1 and a (2)
N is regularly varying with exponent α -1 2 , recall α 2 := min(1 + α, 2)), it is straightforward to see that lim sup

N →∞ P(E + 3 ) ≤ P Y ≥ μ1/α 2 1 2ε
and lim sup

N →∞ P(E - 3 ) ≤ P Y ≤ - μ1/α 2 1 2ε , (3.18)
which are both vanishing as ε 0.

We therefore see that (3.11), (3.15) and (3.18) yield that, provided that ε 0 is small enough, for every ε < ε 0 and N large enough (how large depends on ε), P ε (x) (1η)/μ 1 uniformly for x ∈ {1, . . . , 1/ε}. If we now go back to (3.6) and (3.8), and using that x 1 p h (x) = 1, we obtain(2.4). 3.2. Proof of (2.3). In view of (2.4), we simply need to give an upper bound on the probability P ((N, γ c N + t N ) ∈ τ ). Fix some ε > 0. First step. We control

P M 1,κ N > (1 -ε)t N , τ (1) κ N = N, τ (2) κ N = γ c N + t N ≤ N k=1 k y>-εt N x∈N P τ (2) 1 = t N + y, τ (1) 1 = x P τ (1) k-1 = N -x, τ (2) k-1 = γ c N -y . (3.19)
Recalling (1.11) and (1.12), we have that there is some N 0 = N 0 (ε) and some η = η ε (with η ε → 0 as ε ↓ 0), such that for all N N 0 , x 1/ε and y > -εt N we have

P τ (2) 1 = t N +y, τ (1) 1 = x = L (t N + y + x) (t N + y + x) 2+α e h-xn h (1+η)P τ (2) 1 = t N p h (x), (3.20)
where we recall that p h (x) := (e n h -1)e -xn h . Note that we also have that there is a constant c such that uniformly for x ∈ N

P τ (2) 1 = z, τ (1) 1 = x cL(z)z -(2+α) p h (x) . (3.21)
We can use that to get that uniformly for y t N /2 (so that t N + y t N /2) we have that for any x 1

P τ (2) 1 = t N + y, τ (1) 1 = x c p h (x)P τ (2) 1 = t N . (3.22)
Then, dividing (3.19) according to whether x ≤ 1/ε or x > 1/ε (and summing over y > εt N ), we obtain the following upper bound

(1 + η) 1/ε x=1 p h (x)P τ (2) 1 = t N N k=1 kP τ (1) k-1 = N -x + c N x=1/ε p h (x)P τ (2) 1 = t N N k=1 kP τ (1) k-1 = N -x . (3.23)
The second term is bounded from above by (using also k n)

cN P τ (2) 1 = t N x>1/ε p h (x) = ce -n h /ε × N P τ (2) 1 = t N . (3.24)
In the first term (3.23), we split the sum according to whether k is smaller or greater than (1 + ε)N/μ 1 : we get that

N k=1 kP τ (1) k-1 = N -x ≤ (1 + ε) N μ1 P N -x ∈ τ (1) + N P ∃k > (1 + ε)N/μ 1 s.t. τ (1) k-1 = N -x ≤ (1 + ε) 2 N μ2 1 + N P τ (1) (1+ε)N/μ 1 N -x , (3.25)
where we used that in the first part k (1 + ε)N/μ 1 , and the renewal theorem to get that

P N -x ∈ τ (1)
(1 + ε)N/μ 1 uniformly for x ≤ 1/ε and N large enough (how large depends on ε). The second term is exponentially small since it is a large deviation for τ (1) (x here is bounded by 1/ε). Recalling that p h (x) = 1, the first term (3.23) is therefore bounded from above by

(1 + η) (1 + ε) 2 + e -cεN N μ2 1 P τ (2) 1 = t N , (3.26)
In the end, the left-hand side of (3.19) is bounded by

1 + η ε N μ2 1 P τ (2) 1 = t N with η ε ε→0 → 0 . (3.27)
Second step. It remains to control

P M 1,κ N (1 -ε)t N , τ (1) κ N = N, τ (2) κ N = γ c N + t N = P M 1,κ N ∈ (εt N , (1 -ε)t N ) , τ (1) κ N = N, τ (2) κ N = γ c N + t N (3.28) + P M 1,κ N ≤ εt N , τ (1) κ N = N, τ (2) κ N = γ c N + t N . (3.29)
The first term in the right-hand side, that is (3.28), is smaller than

N k=1 k (1-ε)t N z=εt N x∈N P τ (2) 1 = z, τ (1) 1
= x P τ (2) k-1 = γ c N + t N -z, τ (1) k
-1 = N -x cε -(2+α) N P τ (2) 1 = t N x∈N p h (x) N k=1 P τ (2)
k-1

γ c N + εt N , τ (1) k
-1 = N -x (3.30)
where we used (3.21) uniformly for z εt N and then summed over z to get the first inequality. Then, we split the last sum into two parts. For k k (ε)

N := N/μ 1 + ε 2 t N , we have k (ε) N k=1 P τ (2)
k-1

γ c N + εt N , τ (1) k
-1 = N -x P τ (2) k (ε)
N γ c N + εt N , (3.31)
since the events on the left side are disjoints. Then, provided that ε has been fixed small enough so that

γ c N + εt N μ2 k (ε) N + 1 2 εt N , and since t N /a (2)
N → +∞ (and a

k (ε) N a (2)
N ), we have lim sup

N →∞ P τ (2)
k (ε) N γ c N + εt N = 0 . (3.32)
On the other hand, for k k

(ε) N , we have N k=k (ε) N +1 P τ (1) k-1 = N -x P τ (1) k (ε)
N N -x P τ (1) k (ε)
N μ1 k (ε) N -μ1 ε 2 t N , (3.33)
and since t N / √ N → +∞, also this terms goes to 0 as N → ∞. In the end, we get that the term (3.28) is negligible compared to N P τ (2) 1 = t N . Then, it remains to bound (3.29), and a first observation is that we can restrict it to having κ N k + N := N/μ 1 + t N /(4μ 2 ). Indeed, we have that

P M 1,κ N εt N , τ (1) κ N = N, τ (2) κ N = γ c N + t N , κ N k + N ≤ P τ (1) κ N = N, κ N k + N P τ (1)
k + N N = μ1 k + N -(N -μ1 k + N ) ≤ exp -c (N -μ1 k + N ) 2 /k + N ≤ exp -c (t N ) 2 /N . (3.34)
We mention that the bound for P τ (1)

k + N μ1 k + N -j comes from the fact that -τ (1) 1
is bounded (so it has exponential moments and one can use a Chernov bound, leading to this Gaussian bound). Now, the last term decays faster than N P τ

= t N because of assumption (1. [START_REF] Resnick | A bivariate stable characterization and domains of attraction[END_REF]), provided that C 0 had been chosen large enough.

It remains to control

k + N k=1 P M 1,k εt N , τ (1)
k = N, τ (2)
k = γ c N + t N . (3.35)
We write that each term in the sum is

qn j=log 2 (1/ε) P M 1,k ∈ (2 -(j+1) t N , 2 -j t N ], τ (1)
k = N, τ (2)
k = γ c N + t N , (3.36)
where q N is the smallest integer such that 2 -(q N +1) t N < 1, so q N = O(log 2 N ). Then, using (3.21), each term in the sum (i.e. for every k and j) is bounded by a constant (not depending on j and k) times

k 2 -j t N z=2 -(j+1) t N x∈N L(2 -j t N ) (2 -j t N ) (2+α) p h (x)× P τ (1) k-1 = N -x, τ (2) k-1 = γ c N + t N -z, M 1,k-1 2 -j t N N L(t N ) t 2+α N x∈N p h (x)2 j(3+α) P τ (1) k-1 = N -x, τ (2)
k-1

γ c N + t N 2 , M 1,k-1 2 -j t N , (3.37)
where we used that provided that t N is large enough, L(2 -j t N ) 2 j L(t N ) (this is a direct consequence of Potter's bound for slowly varying functions [3, Th. 1.5.6]) and summed over z. Recovering the sum over k and j, we therefore need to show that

k + N k=1 qn j=log 2 (1/ε) x∈N p h (x)2 j(3+α) P τ (1) k-1 = N -x, τ (2)
k-1

γ c N + t N /2, M 1,k-1 2 -j t N (3.38)
is small for N large. Then, for every j, we define {τ k } k=0,1,... (with distribution P(j) , carrying the dependence on j) as an i.i.d. sum of k variables with distribution (τ

(1) 1 , τ (2) 1 1 {τ (2) 1 2 -j t N } ): we therefore obtain that for k k + N P τ (1) k-1 = N -x, τ (2)
k-1

γ c N + t N /2, M 1,k-1 2 -j t N P(j) τ (1) k-1 = N -x, τ (2)
k-1

γ c N + t N /2 P(j) τ (1) k-1 = N -x, τ (2)
k + N γ c N + t N /2 . (3.39)
Using this inequality and summing it over k in (3.38), (and then using that x p h (x) = 1), we obtain that (3.38) is smaller than

qn j=log 2 (1/ε) 2 j(3+α) P(j) τ (2) k + N μ2 k + N + t N /4 , (3.40)
where we used that γ c N ≥ μ2 k + N -1 4 t N . Then, we may use a Fuk-Nagaev inequality, see for example in [START_REF] Nagaev | Large deviations of sums of independent random variables[END_REF], to control the last probability -we regroup the inequalities we need under the following lemma. Lemma 3.1. Let {X i } i=1,2,... be a sequence of i.i.d. non negative r.v. with P(X 1 > x) ∼ ϕ(x)x -ρ with ρ > 1 and ϕ(•) a slowly varying function. Denote µ := E[X] and σ(y

) = E[X 2 1 1 {X 1 y} ].
We have that there exist constants c, c > 0 such that for any y x

P n i=1 X i 1 {X i y} -µn x 1 + c x nϕ(y)y 1-ρ -cx/y + e -cx 2 /nσ(y) 1 {ρ 2} . (3.41)
Moreover, for any u > 0 there is a constant c u such that, if x/(nϕ(y)y 1-ρ ) u, then

P n i=1 X i 1 {X i y} -µn x exp -c u x 2 nσ(y) . (3.42)
The first term on the right hand side of (3.41) roughly corresponds to the probability to make x/y anomalous jumps of order y, and the second term to a collective Cramér strategy, with no exceptional jump. Lemma 3.1 can be extracted from [START_REF] Nagaev | Large deviations of sums of independent random variables[END_REF]

(y) = E[X 2 1 1 {X 1 y} ] c y 2-ρ ϕ(y), so that log(1 + c x/(nϕ(y)y 1-ρ )) δ u xy/nσ(y) if c x/(nϕ(y)y 1-ρ ) u (this is condition (1.4) in [16, Theorem 1.3]). Applying (3.41) to X 1 = τ (2) 1
(i.e. ρ = 1 + α and ϕ(•) a constant times L(•)), and x = εt N , y = 2 -j t N (so y ≤ x in the summation range for j, cf. (3.40)), we get that (using that σ(y) ≤ σ(x) and k + N N for the term α 1) P(j) τ (2)

k + N µ 2 k + N + t N /4 1 + c ε2 j N P(τ (2) 1 > 2 -j t N ) -cε2 j + e -c ε 2 t 2 N N σ(t N ) 1 {α 1} .
(3.43) The second term poses no issue: summing it for j ∈ [log 2 (1/ε), q n ], we get that

qn j=log 2 (1/ε) 2 j(3+α) e -c t 2 N /N σ(t N ) 1 {α 1} N 3+α e -c t 2 N /N σ(t N ) 1 {α 1} , (3.44)
and this is small when N → +∞ thanks to assumption (1.19), provided that the constant C 0 has been fixed large enough in the case α 1.

For the first term in (3.43), let j ε = j ε (t N ) be the first j such that N P τ

(2) 1 > 2 -j t N c ε A 2 j for some constant A large enough (the precise choice is made below): note that 2 -jε t N is larger than a

N (since N P(τ

(2) 1 > a (2) N ) → 1), so that j ε log 2 (t n /a (2)
N ) → +∞. Summing over j ∈ [log 2 (1/ε), j ε ) the first term in (3.43), we get with the choice c(A -1) 4 + α and ε sufficiently small

jε-1 j=log 2 (1/ε) 2 j(3+α) 1 + ε 1-A -cε2 j jε-1 j=log 2 (1/ε) 2 j(3+α) ε c(A-1)ε2 j ∞ j=0 2 (3+α)(j+log 2 (1/ε))-(4+α)(2 j -1) log 2 (1/ε)-(4+α) log 2 (1/ε) = ε ∞ j=0 2 (3+α)j-(4+α)(2 j -1)
1 = t N )e N n h i 1/ε e -in h K f (i) j 1/ε K f (j) . (2)
Hence, if K = j∈N K f (j) < +∞, we get that provided that ε has been fixed small enough (depending on η), for all N sufficiently large,

II ≥ (1 -η)K N μ2 1 e N n h i≥0 e -in h K f (i) P(τ (2)
1 = t N ) , II ≤ (1 + η)K N μ2 1 e N n h i≥0 e -in h K f (i) P(τ (2)
1 = t N ) . (4.5)
The lower bound (2.7) is obtained simply by using the estimate (2.4) instead of (2.3) in (4.2): the straightforward details are left to the reader.

Analysis of IV and proof of (2.8). It can be written as

IV = i 1/ε t N + 1 ε a (2) N j=t N - 1 ε a (2) N K f (i)K f (j)e (N -i)n h P((N -i, M -j) ∈ τ ). (4.6)
We have that uniformly for j ∈ [t N -1 ε a

(2)

N , t N + 1 ε a (2)
N ], K f (j) ∼ K f (t N ). We can therefore focus on estimating, uniformly for i 1/ε

t N + 1 ε a (2) N j=t N - 1 ε a (2) N P (N -i, M -j) ∈ τ = P for some k, τ (1)
k = N -i, τ (2) k ∈ [γ c N -1 ε a (2) N , γ c N + 1 ε a (2)
N ] ,

and now we prove that this term is close to 1/μ 1 . In fact we have

P for some k, τ (1) k = N -i, τ (2) k ∈ [γ c N -1 ε a (2) N , γ c N + 1 ε a (2) N ] = P(N -i ∈ τ (1) ) -P for some k, τ (1) k = N -i, τ (2)
k / ∈ [γ c N -1 ε a (2) N , γ c N + 1 ε a (2)
N ] ,

and we now show that, provided that ε had been fixed small enough, uniformly for i 1/ε and N large enough:

P τ (1) k = N -i, τ (2) k < γ c N -1 ε a (2)
N for some k ≤ η , (4.7)

P τ (1) k = N -i, τ (2) k > γ c N + 1 ε a (2)
N for some k ≤ η . (4.8) This will be enough, since by the Renewal Theorem we have that P(Ni ∈ τ (1) ) → μ-1

1
uniformly for i 1/ε.

To treat (4.7), define k

N := 1 μ1 N -1 2μ 2 ε a (2)
N : we have uniformly for i ≤ 1/ε P for some k, τ

k = N -i, τ (2) k < γ c N -1 ε a (2) N = P for some k k N , τ (1) k = N -i, τ (2) k < γ c N -1 ε a (2) N + P for some k k N , τ (1) k = N -i, τ (2) k < γ c N -1 ε a (2) N P τ (1)
k N N -1/ε + P τ (2) k N < γ c N -1 ε a (2) N . (4.9) (1)
Now, it is easy to see that the two terms in the last line are small: we indeed have that for arbitrary η > 0, one can choose ε small enough so that for all N large enough,

P τ (1)
k N N -1/ε = P τ (1) k N μ1 k N + μ1 2μ 2 ε a (2)
N -1/ε ≤ η ,

and P τ (2) k N < γ c N - 1 ε a (2) N = P τ (2) k N < μ2 k N - 1 2ε a (2) N ≤ η . (4.10)
For the first line, we used that μ1 2μ 2 ε a

(2)

N -1/ε ≥ ε -1/2 √ N ≥ ε -1/2 √ k N provided
that N is large enough (and ε small), and then simply Chebichev's inequality. For the second line, we used that γ c = μ2 /μ 1 to get that γ

c N = μ2 k N + 1 2ε a (2)
N , and then the approximation of (a

(2) k N ) -1 (τ k N -k N μ2
) by an α 2 -stable distribution, as done in (3.18). For (4.8), we define k

N = 1 μ1 N + 1 2μ 2 ε a (2)
N , and similarly to what is done above, we have

P for some k, τ (1
) k = N -i,τ (2)
k > γ c N + 1 ε a (2) N P τ (1) k N N -i + P τ (2) k N > γ c N + 1 ε a (2) N
, and both terms are smaller than η provided that ε had been fixed small enough and N is large, for the same reasons as in (4.10).

In the end, we get that provided that ε had been fixed small enough, for all sufficiently large

N IV ≥ (1 -η) 1 μ1 K f (t N )e N n h i≤1/ε e -in h K f (i) , IV ≤ (1 + η) 1 μ1 K f (t N )e N n h i≤1/ε e -in h K f (i) . (4.11)
Obviously, since the last sum converges, we can replace it with the infinite sum, and simply replace η by 2η provided that ε is small enough. This competes the analysis of IV.

For what concerns (2.8) we simply need to show that

IVb := Z f N,M,h V (N ) 1 ≤ 1 ε , V (N ) 2 ∈ t N -1 ε a (2)
N , t N + 1 ε a (2) N , M 1,κ N > 1 ε m (2) N = i 1/ε t N + 1 ε a (2) N j=t N - 1 ε a (2) N K f (i)K f (j)e (N -i)n h P (N -i, M -j) ∈ τ , M 1,κ N > 1 ε m (2)
N .

is negligible compared to (4.11). But again, uniformly for the range of j considered, we have K f (j) ≤ 2K f (t N ) (provided that N is large enough). Then, dropping the event Ni ∈ τ (1) , and summing over j, we get that

IVb ≤ 2K f (t N )e N n h i≤1/ε K f (i)e -in h P M 1,κ N > 1 ε m (2) N .
Then, using that κ N ≤ N , we get that

P M 1,κ N > 1 ε m (2) N ≤ P max 1≤k≤N (τ (2)
k - τ (2) k-1 ) > 1 ε m (2) N ≤ N P τ (2) 1 > 1 ε m (2)
N , which can be made arbitrarily small by choosing ε small (uniformly in N ), thanks to the definition (1.16) of m

N . Hence IVb is negligible compared to IV. We also stress here that to estimate IV -in particular to obtain (4.11) -, we did not make use of the assumption K f (i) < +∞.

4.2. Remaining terms. It remains to estimate the terms I, III and V in (4.1), and show that they are negligible compared to (4.4) or (4.11). We start by parts III and V.

Analysis of III. Assume that N is large enough, so that 1 ε a

2) N 1 2 t N we write III Z f N,M,h V (N ) 1 < 1/ε; V (N ) 2 ∈ (1/ε, 1 2 t N ) (denoted IIIa) + Z f N,M,h V (N ) 1 < 1/ε; V (N ) 2 ∈ ( 1 2 t N , t N -1 ε a (
N ) (denoted IIIb) (4.12)

The first term is

IIIa = i<1/ε t N /2 j=1/ε K f (i)K f (j)e (N -i)n h P (N -i, M -j) ∈ τ . (4.13)
Now we can bound, uniformly for i < 1/ε and j t N /2 (so that (M -j)-γ c (N -i) t N /4 for N sufficiently large, and we can apply Theorem 2.1)

P (N -i, M -j) ∈ τ cN sup m t N /4 P(τ (2) 1 = m) c N P(τ (2)
1 = t N ). ( 4

.14)

Hence we get

IIIa ≤ c N P τ (2) 1 = t N e N n h i e -in h K f (i) t N /2 j=1/ε K f (j), (4.15)
and in the case when K f (j) < +∞, the last sum can be made arbitrarily small by choosing ε small. Hence, recalling (4.4), we get that IIIa ≤ η × II for all N sufficiently large, provided that ε is small enough.

For the term IIIb, we use that K f (j) cK f (t N ) for j between t N /2 and t

N -ε -1 a (2) N 2t N (K f (•) is regularly varying), to get that IIIb ≤ i 1/ε t N - 1 ε a (2) N j=t N /2 K f (i)cK f (t N )e (N -i)n h P ((N -i, M -j) ∈ τ ) ≤ c K f (t N )e N n h i 1/ε e -in h K f (i)P for some k, τ (1
) k = N -i, τ (2) k > γ c N + 1 ε a (2)
N .

(4.16)

Since we have seen in (4.8) that the last probability is smaller than some arbitrary η for all N large enough (provided that ε > 0 is small enough), uniformly for all i ≤ 1/ε we have that IIIb ≤ η × IV (recall (4.11)). We stress that, here again, we do not make use of the assumption K f (i) < +∞.

In the end, we obtain that III ≤ η × (II + IV) (provided that N is large enough).

Analysis of V. Let us split V into two parts:

Va = Z f N,M,h V (N ) 1 < 1/ε; V (N ) 2 ∈ (t N + 1 ε a (2) N , 2t N ) , Vb = Z f N,M,h V (N ) 1 < 1/ε; V (N ) 2 > 2t N .
Let us start with Va. We proceed analogously as above: using that K f (j) cK f (t N ) uniformly for j ∈ (t N , 2t N ), we get that Va

i 1/ε j t N + 1 ε a (2) N K f (i)cK f (t N )e (N -i)n h P (N -i, M -j) ∈ τ cK f (t N )e N n h i 1/ε e -in h K f (i)P for some k, τ (1)
k = N -i, τ (2) k < γ c N -1 ε a (2)
N .

Now we again recall (4.7), which tells that the last probability is smaller than some arbitrary η for all N large enough (provided that ε > 0 is small enough, uniformly for all i ≤ 1/ε). In the end, in view of (4.11), we get that Va ≤ η × IV, and here again we did not make use of the assumption K f (i) < +∞. For the term Vb, we write Vb = i 1/ε j 2t N K f (i)K f (j)e (N -i)n h P (Ni, Mj) ∈ τ .

(4.17)

Then, we estimate the last probability: by setting k N := 1 μ1 N -1 2μ 2 j (for some small c 0 ), we obtain as in (4.9)-(4.10) that P (Ni, Mj) ∈ τ P τ (1)

k N N -1/ε + P τ (1) k N γ c N -j P τ (1)
k N μ1 k N + μ1 2μ 2 j -1/ε + P τ (2)
k N μ2 k Nj/2 . (4.18)

Then, the two terms are bounded by e -cj 2 /k N : recall that τ (1) has an exponential tail, and that -τ (2) is bounded above, so that they have some exponential moments (and we For the second term, similarly to term V, we split it according to whether V K f (i)K f (j)e (N -i)n h P (Ni, Mj) ∈ τ

cK f (t N )e N n h i≥1/ε K f (i)e -in h , (4.23)
where we used that K f (j) cK f (t N ) for j ∈ (t N /2, 2t N ), and that the sum over j of P (Ni, Mj) ∈ τ is bounded by 1. Then, the last sum can be made arbitrarily small by choosing ε small, so that in view of (4.11), this term can be bounded by η × IV.

The second contribution is

Z f N,M,h V (N ) 1 ∈ (1/ε, (log N ) 2 ), V (N ) 2 > 2t N = i 1/ε M j=2t N K f (i)K f (j)e (N -i)n h P (N -i, M -j) ∈ τ Ce N n h M j=2t N K f (j)e -cj 2 /k N Ce N n h N c+1 e -ct 2 N /N , (4.24)
where we used (4.18), and (4.19). Then, this term decays faster than IV, provided that C 0 has been fixed large in (1.19), as for the term Vb.

Conclusion in the case of j K f (j) < ∞. We have therefore proven that for any η > 0, we can choose ε > 0 small such that, for all N large enough (how large depend on ε), II + IV ≤ Z f N,M,h ≤ (1 + 2η)II + (1 + 4η)IV (4.25)

and the two terms behave asymptotically respectively as (4.4) and (4.11): this proves (2.6) for j K f (j) < ∞.

The case of j K f (j) = ∞, with α < 1. This time we have to show that IV dominates. We go through the various terms, but as pointed out during the proof, we have not used j K f (j) < ∞ in estimating IV, so (4.11) still holds. We retain, for local use, that IV behaves (and, in particular, is bounded from below by) a constant times K f (t N ) exp(N n h ).

The estimate (4.4) for II is still valid. This term can be dealt directly without troubles, but it is more practical to observe that II (and the right side of (4.15), which bounds IIIa) is bounded (for N large) by a constant times Plugging t N = a √ N log N in (B.5), we find that the first term is regularly varying with index -α/2 and that the second term has index -1/2ca 2 . Hence, depending on a we can identify the dominant term in (B.5):

N P τ (2) 1 = t N exp(N n h ) j≤t N /2 K f (j) . ( 4
1 st term is dominant if a > α -1 2c
and 2 nd term is dominant if a < α -1 2c .

We therefore interpret this as a c = (α -1)/2c, where a c is the critical value mentioned at the beginning of Appendix B.

t N = (γ -γ c )N t N = (γ -γ c )N t N = O(a (2)
N ) with the different regimes. We treat the big-jump domain and it is the one to the right of the right-most dashed line. We believe that to the right of the moderate Cramér regime there is the big-jump domain -put otherwise, the non-Cramér regime coincides with the big-jump domain -but this is proven only for α < 1.

-t N ≪ N -t N ≪ N t N ≫ a (2) N γ < γ c γ < γ c t N < a c √ N log N t N > C 0 √ N log N big-jump Conjecture: a c = C 0
Notice that, when t N / √ N → -∞, one could develop an identical argument (except that the big-jump term disappears), provided that a local moderate deviation theorem as (B.4) holds -i.e. provided that |t N |/ √ N is not too large, how large depend mostly on the tail exponent 1 + α > 2 of τ (2) . In the end, the sharp asymptotics of P (N, γ c N + t N ) ∈ τ should also be given by the second term in (B.5) -as already seen in the case t N / √ N → t ∈ R in Appendix B.1.

Figure 1 .

 1 Figure1. A configuration of the free gPS model, with one strand containing 22 bases and the other 14, is represented in two fashions: the natural (or polymer) one and the renewal one. In particular we see that (1, 1) renewal increments (or inter-arrivals) correspond to bound base pairs and all other increments (i, j) correspond to unbound regions in the bulk, that we call loops (of total length i + j, with length i in the first strand and j in the second strand). The term unbound is rather reserved to the terminal portion of the polymer: we refer to the free ends as unbound strands. Throughout this work, a polymer trajectory is always given in the renewal representation: it is therefore just a point process in the plane.

  Fig 2.

Figure 2 .

 2 Figure2. A schematic image of the two types of observable trajectories of the free gPS model in the delocalized (denaturated, melted) regime, according to whether the exponent α is smaller (left picture) or larger (right picture) than 1 + α/2. In the constrained case only the trajectory on the right is observed, and the small free tails are reduced to zero. This case is treated in[START_REF] Giacomin | Generalized Poland-Sheraga denaturation model and two dimensional renewal processes[END_REF].

Figure 3 .

 3 Figure 3. Schematic image of the observable trajectories of the free gPS model in the Cramér regime (left), and in the non-Cramér regime (cf. Theorem 1.1): the Large Unbound Strand event (center, occuring when α < α + 1) and the Big Loop event (right, occuring when α > α + 1). In the constrained case the Unbound Strand event is not observed, and the free tails are of course absent. What cannot be appreciated in this schematic view is the fact that the small loop distribution has exponential tail in the Cramér regime (hence the largest is O(log N )) and that it has power law tail in the non-Cramér regime (hence the largest is O(N a ) for some a ∈ (0, 1): O(mN ) to be precise).

  log 2 (1/ε) 2ε . (3.45) (Recall t n := Mγ c N and (1.12).) Hence,

(N ) 2 ∈ 1 ∈

 21 (t N /2, 2t N ) or V (1/ε, (log N ) 2 ), V

Figure 4 .

 4 Figure 4. A schematic sum-up of correspondence of the the values of tN = (γ-γc)N with the different regimes. We treat the big-jump domain and it is the one to the right of the right-most dashed line. We believe that to the right of the moderate Cramér regime there is the big-jump domain -put otherwise, the non-Cramér regime coincides with the big-jump domain -but this is proven only for α < 1.

  2, separating a big-jump domain (when t N > a √ N log N with a > a c ) from a moderate Cramér regime (when t N < a √ N log N with a < a c ).

	α < 1	Cramér	moderate Cramér	non-Cramér
	α > 1	Cramér	moderate Cramér	non-Cramér

Acknowledgements

G.G. acknowledges the support of grant ANR-15-CE40-0020.

For j j ε , we use (3.42). In the case α 1, we therefore have the same reasoning as in (3.44): the sum for j ∈ [j ε , q n ] is small. If α < 1, then σ(y) = E[X 2 1 1 {X 1 y} ] cy 2 ϕ(y)y -ρ , and (3.43) becomes, for j j ε P(j) τ (2)

where we used in the second inequality that (N P(τ

1 > 2 -j t N )) -1 is bounded above by a constant if j ≥ j ε . Then, summing over j ∈ [j ε , q N ], we get q N j=jε(t N ) 2 j(3+α) e -c ε 2 2j → 0 as N → +∞.

(3.46)

All together, we have shown that (3.40) is small.

4. The free partition function: proof of Theorem 2.2

We will first prove (2.6) for the case j K f (j) < ∞. Many estimates are in common with the case j K f (j) = ∞ that we treat right after, and we will stress along the proof when the estimates are dependent or not on the fact that j K f (j) < ∞. Also, the proof of the lower bounds (2.7) and (2.8) are contained in the proof of (2.6) as we explain along the way. Proof of Theorem 2.2. As announced, we start with the proof of (2.6) and assume j K f (j) < ∞. Let us fix η > 0, and ε > 0 small, how small depends on η as will be stressed in the proof.

We decompose the free partition function into several parts:

N .

(4.1)

The main contribution comes from the terms II and IV. We first estimate these terms, before showing that all the other ones are negligible compared to max(II, IV).

4.1.

Main terms, and proof of (2.7) and (2.8).

Analysis of II and proof of (2.7). Recall (1.3) and (1.10). The term II can be written as

and it is just a matter of estimating P((Ni, Mj) ∈ τ ) uniformly for 0 i, j 1/ε. We have from Theorem 2.1, uniformly for i, j 1/ε,

may use standard Chernov bounds, as in (3.34)). All together, we get that

where we used that K f (j) is bounded by N c for all j M , for some c > 0. Then, if α < 1 then there is a δ > 0 such that t N a

(2) N N 1/2+δ , so that exp(-c(t N ) 2 /N ) decays faster than any power of N . In the case α 1, we get thanks to (1.19) that exp(-c(t N ) 2 /N ) decays faster than N -c C 0 : hence choosing C 0 arbitrarily large, we get that (4. [START_REF] Resnick | A bivariate stable characterization and domains of attraction[END_REF]) is negligible compared to IV.

Analysis of I.

We separate it into two parts:

We have

where we first simply bounded the probability by 1, and also that there is some constant c > 0 such that K f (i) ≤ cN c for i ≤ N, M . Clearly, in view of (4.11) (or (4.4)), we get that Z f N,M,h (V (N ) 1

(log N ) 2 ) = o(IV), since 1/K f (t N ) and 1/P(τ

) for some c > 0. Again, we did not use that K f (i) < +∞, even if it would have simplified the upper bound.

We now turn to the case when V (N ) 1 ≤ (log N ) 2 . We write

For the first term, and using that

When j K f (j) < +∞, then recalling (4.4), this term is smaller than η × II provided that ε is small enough.

N . This shows that both II and of the right side of (4.15) (which bounds IIIa) are negligible compared to IV.

The estimates for IIIb and V, as already pointed out, are valid without assuming that j K f (j) < ∞, so we are left with controlling I. Recall that we split the contribution of I into three parts: (4.20), (4.22) and (4.23). As noticed above, the fact that j K f (j) < ∞ was not used in estimating (4.20) and (4.23). Moreover (4.22) we can be bounded like (4.15) (in fact, it is much smaller), that was found above to be negligible compared to IV. We therefore conclude that I is also negligible compared to IV, and this completes the analysis of the case j K f (j) = ∞, and of the proof of Theorem 2.2.

Appendix A. The case α = 1 and j K f (j) = +∞ Recall the definition (2.9) of the Big Loop and Unbound strand, and when α = 1 with

The event E

(N ) mixed is therefore a set of trajectories with both a big loop (of order t N ), and a large unbound strand (of large order, but much smaller than t N ) -to optimize the interval for the length of the unbound strand, one can take v N → +∞ and ε N → 0 as fast as possible, with the limitation given by (A.2).

Theorem A.1. Suppose that α > 0 and (1.18), and if α 1 assume additionally (1.19). We assume that α = 1 and that j K f (j) = +∞, and we denote K(x

Obviously, this theorem can easily be translated in term of path properties. Indeed, since P(τ

1 > t N ) and K f (t N ) = L(t N )t -1 N , we have the asymptotic of the ratio

with K(x)/L(x) → +∞ as a slowly varying function. Therefore, we obtain

We stress that, when α > 1, the ratio Q N always goes to 0 as N → ∞: indeed, in that case N P(τ

1 > t N ) decays faster than any slowly varying function. However, in the case α ∈ (0, 1], the ratio Q N diverges when t N → +∞ slowly enough, showing that there is a regime under which the mixed trajectories described in the event E (N ) mixed occur, in the sense that P f N,M,h (E

Proof. We treat this case in a concise way because most of the technical work has already been done above. To summarize -recall the different contributions in (4.1) -the term IV is well estimated in (4.11) and the terms I, IIIb and V were found to be negligible compared to it -this was valid even when j K f (j) = +∞. When j K f (j) = +∞, then the term II is found to be negligible compared to the right side of (4.15) (which bounds IIIa), and we therefore focus on this last term. We can again decompose IIIa into two contributions:

The second one, exactly in the same manner as for IIIb, can be shown to be negligible compared to IV as N → ∞. Then, the first term is equal to

Then, thanks to Theorem 2.1, for every η > 0 we can choose ε > 0 small enough and N ε large enough so that uniformly for the range of i and j considered, and

and we stress that the main contribution to this probability comes from a big loop event, of length larger than (1ε)t N . We therefore get that, for N large enough, and denoting K(x) := x j=1 K f (j) which is a slowly varying function,

and similarly for an upper bound with 1η replaced by 1 + η.

We are actually able to narrow the condition V (N ) 2

∈ (1/ε, εt N ) in IIIa to a smaller interval (v N , ε N t N ) without changing the estimates, provided that v N → +∞ and ε N → 0 slowly enough, so that condition (A.2) holds.

Appendix B. About the transition between Cramér and non-Cramér regimes

In this Appendix, we discuss the condition (1.18)- (1.19) ensuring that one lies in the bigjump regime described by Theorem 2.1. We focus on the constrained partition functionor rather the probability P((N, M ) ∈ τ ) -to study the transition between the condensation phenomenon that we highlighted and the Cramér regime, but all the observations made here could also apply to the other results. Like in Section 4 we omit integer parts, so γ c N stands for the (upper or lower, as one wishes) integer part of γ c N . B.1. Between Cramér and non-Cramér regimes I. If one sets M = γ c N , or in other words if t N = 0, then [START_REF] Berger | Strong renewal theorems and local large deviations for multivariate random walks and renewals[END_REF] proves that

where the constant c 0 > 0 is explicit. The heuristics of this result can be easily understood: the typical number of renewal is k N = N/μ 1 + O( √ N ) and, for each k in that range, Doney's Local Limit Theorem [START_REF] Doney | A bivariate Local Limit Theorem[END_REF] gives that P(τ k = (N, M )) is equivalent up to a multiplicative constant to (a

Hence, neither τ (1) nor τ (2) have to make an atypical deviation, and the term (a

N ) -1 simply comes from a local limit theorem: there is no condensation phenomenon, i.e. the typical trajectories contributing to the event (N, γ c N ) ∈ τ do not exhibit a big jump. However, we are not in the Cramér regime -one component of the inter-arrivals does not have exponential tails, so there are jumps that are luch larger than log N -and we can see this critical situation as a moderate Cramér regime, because (moderate) deviations are carried by both components, like in the Cramér regime the (large) deviations are carried by both components.

A behavior like (B.1) also holds when t N /a

N → t ∈ R: the constant c 0 is simply replaced by a constant c t depending on t. When α ∈ (0, 1), the fact that one lies in the big-jump regime (and Theorem 2.1 holds) as soon as t N /a

(2) N → +∞ is optimal, in the sense that when sup N t N /a

(2) N < +∞, then the typical trajectories do not exhibit a condensation phenomenon. B.2. Between Cramér and non-Cramér regimes II. When α ≥ 1, the situation is more involved because the condition t N /a

N → +∞ alone is not enough to ensure that the model is in the big-jump domain.

We conjecture that when α > 1, there is some a c = a c (α) -that we give explicitly below -such that the big-jump regime holds when t N > a √ N log N with a > a c (i.e. theorem 2.1 holds), and a moderate Cramér regime holds when t N < a √ N log N with a < a c (we give an explicit conjectured analogue of Theorem 2.1, see (B.5) below). Note that having t N > a √ N log N corresponds to our condition (1.19), except that our constant C 0 is not optimal. Finding the correct threshold when α = 1 is even more involved and we prefer to leave it aside.

So let us now focus on the case α > 1, and develop some heuristic arguments to conjecture the asymptotic behavior of P (N, M ) ∈ τ , and the typical behavior of trajectories contributing to this event. We take a

should not be excluded), in which case the cost is shared jointly by the two coordinates τ (1) , τ [START_REF] Berger | Strong renewal theorems and local large deviations for multivariate random walks and renewals[END_REF] .

Then, having a look at Nagaev's Theorem 1.9 in [START_REF] Nagaev | Large deviations of sums of independent random variables[END_REF] suggests that for any k, only two possible behavior can contribute to P(τ

k = γ c N + t N ): having one large jump (in which case, and since τ (2) has a heavier tail, the probability is maximal when k = N/μ 1 + O( √ N ) so that only τ (2) has to make a large jump), or using a collective joint strategy with no big jump (i.e. a moderate Cramér regime). The first possible behavior is therefore the big-jump strategy that we already identified, and we would therefore have that

where by "no big jump" we mean that all jumps are O(m N ). Using a local moderate deviation theorem for the probability when no big jump occurs (such a local moderate deviation theorem should hold because t N is not too large, t N ≤ C 0 √ N log N ), we would have that, for k = N/μ 1 + θt N /μ 2 with c := μ1 2 (γ 2 c σ 2 1 + 2ργ c σ 1 σ 2 + σ 2 2 ) -1 , and the constant c 1 could in principle be made explicit.


Matthias Heinrich
email: matthias.heinrich@enpc.fr

Latts

Cosys-Grettia

Marie Ruellan
email: marie.ruellan@cyu.fr

Jean-Pierre Lévy
email: jean-pierre.levy@enpc.fr

Allou Samé
email: allou.same@univ-eiffel.fr

Latifa Oukhellou
email: latifa.oukhellou@univ-eiffel.fr

What insights can we draw from our residential energy models? Guidelines for future modelling exercises

Keywords: Buildings, Energy model, Heuristics, XGBoost, Prediction

Understanding energy consumption in the residential sector remains challenging. In current regression models, prediction performance on a test dataset is frequently used to justify the ability of the associated model to correctly describe the energy consumption. However, prediction and explanation are often conflicting goals in modelling and most models convey heuristics rather than a precise representation of the processes involved. From this point of view, comparing and analyzing several modelling strategies is a way to improve our comprehension. In this work, we studied both the prediction performance and the heuristics provided by nine models of housing energy consumption built from survey data of 2000 French households. Concerning prediction, the eXtreme Gradient Boosting algorithm based on features extracted from surveys provided the best predictor with an average MAPE of 38%. Residual analysis showed that the models generally make the same type of errors, with absolute errors being much larger for houses than for apartments. This suggests that the improvement in predictive performance relies more on the introduction of supplementary predictive variables than on increasing the model's complexity. In terms of interpretation, all models identify building type, floor area, household composition and heating energy as key determinants of final energy consumption. We illustrate, through t-SNE mapping, that these variables are in fact very much interrelated. The study therefore underlines the high interest of studying the interaction between the variables. Explanatory and predictive models could therefore be based on an adapted segmentation of dwellings and households. This type of modelling could also join the work in the sociology of energy which largely describes the interweaving of these variables within lifestyles.

Introduction 1.Problem & context

The building sector accounts for nearly 40% of global energy consumption and 20% of greenhouse gas emissions and, as stipulated by the Paris Agreement, this sector must implement drastic reductions in consumption [START_REF] Lane | IEA Energy Efficiency 2018 and World Energy Outlook[END_REF]. Numerical modelling is a major tool in this sector as it allows one to better understand and quantify the effect of different parameters such as insulation, inhabitants' behaviours, climate, and equipment [START_REF] Emery | A long term study of residential home heating consumption and the effect of occupant behavior on homes in the Pacific Northwest constructed according to improved thermal standards[END_REF]- [START_REF] Foucquier | State of the art in building modelling and energy performances prediction: A review[END_REF]. However, models have difficulty in estimating energy consumption accurately -a phenomenon known as the energy performance gap [START_REF] Zou | Review of 10 years research on building energy performance gap: Life-cycle and stakeholder perspectives[END_REF], which reflects a limited understanding of the processes at work, limited data, and imperfect models. As a result, there is a large body of literature in various disciplines including data-science, engineering, anthropology and sociology at all spatial and temporal scales [START_REF] Zou | Review of 10 years research on building energy performance gap: Life-cycle and stakeholder perspectives[END_REF], [START_REF] Amasyali | A review of data-driven building energy consumption prediction studies[END_REF].

Many physical and data-based numerical models have been developed. The purpose can be to predict residential consumption at a given spatial and temporal scale [START_REF] Foucquier | State of the art in building modelling and energy performances prediction: A review[END_REF], [START_REF] Amasyali | A review of data-driven building energy consumption prediction studies[END_REF], or to propose an explanation by inferring an understanding of the phenomenon [START_REF] Aydinalp-Koksal | Comparison of neural network, conditional demand analysis, and engineering approaches for modeling end-use energy consumption in the residential sector[END_REF]- [START_REF] Estiri | Age matters: Ageing and household energy demand in the United States[END_REF]. Several reviews have been carried out in recent years [START_REF] Foucquier | State of the art in building modelling and energy performances prediction: A review[END_REF], [START_REF] Amasyali | A review of data-driven building energy consumption prediction studies[END_REF], [START_REF] Zhao | A review on the prediction of building energy consumption[END_REF], [START_REF] Liu | Accuracy analyses and model comparison of machine learning adopted in building energy consumption prediction[END_REF] to assess the state-of-the-art in the construction of predictive models. These studies divide the models into 3 categories: engineering methods such as dynamic thermal simulation, statistical methods such as multiple linear regression (MLR), conditional demand analysis, and artificial intelligence methods such as artificial neural networks [START_REF] Ahmad | Trees vs Neurons: Comparison between random forest and ANN for high-resolution prediction of building energy consumption[END_REF], support vector machine [START_REF] Ma | Applying support vector machines to predict building energy consumption in China[END_REF], random forest [START_REF] Ma | Identifying the influential features on the regional energy use intensity of residential buildings based on Random Forests[END_REF], or extreme gradient boosting (XGB) [START_REF] Le | Estimating the Heating Load of Buildings for Smart City Planning Using a Novel Artificial Intelligence Technique PSO-XGBoost[END_REF]. The review papers point out that a minority of the work reviewed has focused on residential buildings notably because of data availability. A large part of the literature focuses on short term prediction based on sensor data [START_REF] Kong | Short-Term Residential Load Forecasting Based on LSTM Recurrent Neural Network[END_REF]. Other papers mobilize building, household and behaviour data to model long term energy forecasts [START_REF] Emery | A long term study of residential home heating consumption and the effect of occupant behavior on homes in the Pacific Northwest constructed according to improved thermal standards[END_REF], [START_REF] Allibe | Modélisation des consommations d'énergie du secteur résidentiel français à long terme -Amélioration du réalisme comportemental et scénarios volontaristes[END_REF]. These can either focus on consumption related to end uses, such as heating, cooling, ventilation, or lighting, or on a type of energy (electricity, wood, gas, fuel oil, etc.). The investigations discuss architecture selection and dimension reduction through feature extraction strategies to improve prediction: they apply linear techniques such as factor analysis, or non-linear techniques such as auto-encoding using neural networks. As stated above, these predictive models suffer from considerable imprecision, which is greater for predictive models with a short time horizon (hour, day), or reduced spatial resolution compared to annual models or models at a higher spatial scale. In general, machine learning modelling provides better predictors than engineering modelling.

These models are often used also to provide explanations about energy consumption in dwellings. Regression models are widely used to identify and quantify the causal effects of explanatory variables such as household, housing, and behavioural characteristics on energy consumption [START_REF] Emery | A long term study of residential home heating consumption and the effect of occupant behavior on homes in the Pacific Northwest constructed according to improved thermal standards[END_REF], [START_REF] Belaïd | Untangling the complexity of the direct and indirect determinants of the residential energy consumption in France: Quantitative analysis using a structural equation modeling approach[END_REF]- [START_REF] Guerra Santin | The effect of occupancy and building characteristics on energy use for space and water heating in Dutch residential stock[END_REF] because they generally provide a good predictive performance, thus making validation possible, and because they carry with them a strong theoretical framework based on the additivity of individual effects "all other things being equal" [START_REF] Shmueli | To Explain or to Predict?[END_REF]. An example is the work of Belaïd [START_REF] Belaïd | Untangling the complexity of the direct and indirect determinants of the residential energy consumption in France: Quantitative analysis using a structural equation modeling approach[END_REF] who used a structural equation modelling approach to estimate the energy consumption of 2356 French households. To do so, he used data from the 2013 PHEBUS survey conducted by the French Ministry of Ecology. The data describe households and dwellings, behaviours, and final energy consumption. After validating the model, the article quantified the direct and indirect effects of household and housing characteristics on their energy consumption to support proposals for public action, in particular the thermal renovation of housing, which is shown to have a greater impact than household characteristics. Machine learning models such as decision trees and ensemble models can also provide useful information [START_REF] Breiman | Random Forests[END_REF]. Decision trees provide a set of rules based on variables and thresholds. The variables used in ensemble models such as random forest and XGB can be compared in terms of their contribution to the accuracy of the algorithm.

However, it is important to mention that there is no consensus in the scientific community about the interest of validating or not the predictive performances of the models before drawing any conclusions from them. Shmueli [START_REF] Shmueli | To Explain or to Predict?[END_REF] reminds us that predicting and explaining are two different goals that should not be confused. He shows that it is sometimes possible to build a "false" model capable of providing better predictions than the "true" model. Furthermore, in a seminal article published in Science in 1994, Oreskes [START_REF] Oreskes | Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences[END_REF] recalled that: "Models can only be evaluated in relative terms, and their predictive value is always open to question. The primary value of models is heuristic." In this perspective, the predictive performance of the models occupies a more nuanced place: rather than simply justifying an interpretation or validating a set of processes, it informs on the model's capacity to reproduce the correlations through a set of heuristics. At this point, it is possible to formulate two different questions for these models: what are the most suitable modelling structures and features to build a good predictor? What insights are provided by models of diverse complexity?

We propose in this work first to compare both the predictive and explanation performance from models classically used in the literature. This includes the comparison of 3 models of different complexity (decision tree, multiple linear regression, XGB model) and 3 feature extraction strategies (raw variables, principal components, and variables from autoencoder training). Then, by analysing these results, confronting them with a t-SNE observation of the data and by mobilising the literature in the sociology of energy, we try to justify the interest of a segment-based approach to build more accurate and realistic models.

Methodology, data, and models

The methodology followed is summarised in the figure below.

Figure 1: Methodology of this work

The study uses data from the 2013 PHEBUS survey conducted by the statistical department of the current French Ministry of Ecological Transition [START_REF]Enquête Performance de l'Habitat, Équipements, Besoins et Usages de l'énergie (Phébus) | Données et études statistiques[END_REF]. The sample consists of 8000 dwellings representative of French regions, climatic zones, housing types and years of construction. The data collected describe the occupied dwellings, including heating and renovations that have been carried out, the socio-demographic characteristics of households, energy consumption and energy behaviour. We used 47 variables from this survey, 11 of which describe households (income, sparings, composition, age of the reference person, number of person and consumption units, socio-professional group, …), 1 final energy consumption (noted FEC, as Final Energy Consumption), 12 dwellings (type, surface, number of rooms and bedrooms, occupancy status, date of construction, urban area, insulation work, Unified Degree Days, type of heater and age of heater, …) and 23 residential behaviour (including average attendance, number of equipment's, declared heating temperature, number of showers per day, heating regulation level, heating deprivation,…). The dataset comprises 1999 households for which the data are complete. The data used are not freely available but can be requested with the agreement of the producer and the consent of a statistical confidentiality committee. In this modelling work the dependent variable is the logarithm of final energy consumption. The numerical variables are standardised and centred. In this study we built a total of 9 final energy consumption models. For that purpose, we combined 3 feature extraction strategies (Simple Features "SF", Principal Components "PC", and Autoencoding "AE") and 3 model frameworks (decision tree "DT" [START_REF] Breiman | Random Forests[END_REF], multiple linear regression model "MLR" [START_REF] Breiman | Random Forests[END_REF], eXtreme Gradient Boosting "XGB" [START_REF] Chen | XGBoost: A Scalable Tree Boosting System[END_REF]). A few details on the features extraction strategies are given here. The SF are coded as dummy variables for the multiple linear regression (MLR) and by one hot coding (OHE) for the XGB training. PC feature extraction uses Factor Analysis on Mixed Data (FAMD) [START_REF] Pagès | Analyse factorielle multiple appliquée aux variables qualitatives et aux données mixtes[END_REF]. By performing three FAMDs on the groups of household, housing, and energy behaviour variables, we generated a total of 52 synthetic variables to capture 80% of the variance each time. The calculations were performed on R using the FactoMineR library. The AE variables were generated by neural networks called autoencoders. As the data are sparse, we used a cosine loss function [START_REF] Tensorflow | Keras Losses -CosineSimilarity[END_REF]. The size of the hidden layer was chosen to be 15 to achieve a threshold of -0.85 of the loss function. The encoding calculations were performed in Python using the Keras library [START_REF] Chollet | [END_REF]

Model performance 2.1 Prediction performance

Each model was trained on 80% of the database, i.e. 1600 households. They were then evaluated on their predictive performance, i.e. their ability to predict the annual final energy consumption of the 20% of households not included in the training set. To reduce the dependence of this evaluation on data selection, the models were trained and evaluated 50 times on randomly drawn samples. 4 indicators were calculated from the predictions of the models (R², Root Mean Square Error "RMSE" in kWh, Mean Absolute Error "MAE" in kWh, Mean Absolute Percentage Error "MAPE" in percent). The average of these 4 indicators over the 50 trainings are given in Table 1. We used the caret library implemented in R. As a result, the choice of the metric has little or no impact on the ranking of the models. We see that the XGB-SF model offers the best predictive performance, slightly ahead of the MLR-SF multiple linear model. Secondly, we observe that feature extraction and the resulting loss of information degrades the predictive performance of the models.

Model interpretation

The analysis of the variables selected by the models generally makes it possible to draw insights about the phenomenon which is modelled. After computing several models, we noticed that with two notable exceptions (the variables describing the amount of sparings and the declared heating temperature) all the variables selected by DT were also selected by the XGB model. And all the variables selected by XGB were also selected by the MLR model.

Whatever the feature extraction strategy and the model, the variables describing the dwelling, the heating energy, and the size of the dwelling were selected first. This confirms the important role of building characteristics on the FEC. However, the variables characterising the household and domestic behaviours occupy higher or lower ranks, depending on the training sample, but also on the feature extraction strategy. We compared the residuals of the MLR-SF, DT-SF and XGB-SF models to observe whether the modelling structure allows for a change in the nature of the errors or whether, on the contrary, all the errors are the same, which may then highlight a problem with the data or the modelling assumptions made. We observed that the residuals of the three models were similar and that the models tend to underestimate (resp. overestimate) low (resp. high) consumption levels.

Conclusion and key messages : How to design more efficient energy models?

Modellers would like to be able to provide a generic answer to explain and predict housing energy consumption through the abstractions and variables of the physical and social phenomena that models carry out. Obviously, the performance of current predictive models is modest as underlined by the indicators of the best predictor XGB-SF: R² ~ 63.1%, RMSE ~ 6.8 MWh, MAE ~ 4.4 MWh, MAPE ~ 37.7% and the modelling interpretation says not much more than that the characteristics of the dwelling are determinant. Depending on the target, we would consider two key messages for future energy modelling exercises.

For prediction: search for more (and better) data. We found in this work that making the features and structure of the regression model more complex brought relatively little prediction performance. Furthermore, the errors in the models were of the same type, which suggests that the "raw" variables are good but that some are missing or that the data were not good enough.

For explanation: Consider the significant correlations between the variables. The calculation of independence tests between the characteristic variables of the household (composition of the household, age of the reference person, occupation status), the dwelling (surface, type, type of heating) and the domestic behaviours give very significant p-values (<5%). To observe these correlations, we project the households and dwellings in a twodimensional space. The space is constructed by t-SNE (see Maaten and Hinton [START_REF] Van Der Maaten | Visualizing Data using t-SNE[END_REF]) on the variables after one-hot encoding with a perplexity parameter of 50, which provided stable and relevant results. Note that the criterion for constructing the space is the relative position of individuals, which means that the axes cannot be interpreted with the initial variables. In figure 2, we have plotted the households map by intersecting it with variables such as surface area, type of dwelling, household composition, logarithm of final energy consumption, number of persons. The variable FEC was not used to create this synthetic space. Several remarks can be made. Firstly, the map allows several distinct and homogeneous zones to be distinguished for each variable. For example, the map describing the housing type highlights the fact that flats are mainly located on the left while single-family houses are mainly on the right. Additionally, regarding household composition, we can see that families with children are present in two distinct blocks on the right and left of the graph (in fact, it is the most affluent households on the right and the most deprived households on the left). Single people are positioned in the center while couples without children are located on the lower part. When plotting the maps by highlighting the level of energy consumption, we can see a strong separation between the less energy consuming flats and the more energy consuming detached houses. The visualization is interesting because it allows for going beyond a simple correlation and independence analysis: in particular, it suggests exploring a database segmentation exercise. This approach would make it possible to carry out separate models for flats and single-family houses, since the dynamics in terms of consumption seem to be very different. Also, modelling by segment would make it possible to account for the recurrent associations between the modalities of the different variables observed here in the data (e.g. couples without children living in a small -most likely old -gas-heated detached house). These associations are consistent with results from the sociology of energy, which describes the association between types of household, types of housing, types of energy [START_REF] Lévy | The determinants of domestic energy consumption in France: Energy modes, habitat, households and life cycles[END_REF] and behaviour [START_REF] Heinrich | From energy behaviours to lifestyles: contribution of behavioural archetypes to the description of energy consumption patterns in the residential sector[END_REF]. We believe that it is not certain that segmentbased modelling will significantly improve the predictive performance of housing energy consumption models. However, segment-based models make it possible, if not to quantify precisely, to evaluate in relative terms the effect of any changes in this segment (insulation, regulation behaviour, energy efficiency of equipment, etc.).

ACKNOWLEDGMENTS

This work was partially funded by the French Ministry of Ecological Transition. We would like to thank the reviewers and the editor for their comments and suggestions which allowed to significantly improve the article.

Figure 2 :

 2 Figure 2: t-SNE visualization of household, housing, and behaviour variables. The final energy consumption variable (FEC) is not used in the calculation. (Data: Secondary processing from PHEBUS data). Abbreviations: AB: Apartment Block; TH: Terraced House; MFH: Multi-Family house; SFH: Single Family House

Table 1 : Average prediction performance for all models (DT, MLR, XGB) and feature extraction strategies (SF, PC, AE). The average indicators are calculated from 50 runs with random selection of test datasets. Top scores per column are shown in bold. (Data: Secondary processing from PHEBUS data).

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