Quentin Berger
email: quentin.berger@sorbonne-universite.fr

Niccol Ò Torri
email: niccolo.torri@sorbonne-universite.fr

DIRECTED POLYMERS IN HEAVY-TAIL RANDOM ENVIRONMENT

Keywords: 2010 Mathematics Subject Classification: Primary: 60F05, 82D60; Secondary:60K37, 60G70 Directed polymer, Heavy-tail distributions, Weak-coupling limit, Last Passage Percolation, Super-diffusivity

.

We consider the directed polymer model: it has been introduced by Huse and Henley [START_REF] Huse | Pinning and roughening of domain walls in ising systems due to random impurities[END_REF] as an effective model for an interface in the Ising model with random interactions, and is now used to describe a stretched polymer interacting with an inhomogeneous solvent.

Let S be a nearest-neighbor simple symmetric random walk on Z d , d ě 1, whose law is denoted by P, and let pω i,x q iPN, xPZ d be a field of i.i.d. random variables (the environment) with law P (ω will denote a random variable which has the common distribution of the ω i,x ). The directed random walk pi, S i q iPN 0 represents a polymer trajectory and interacts with its environment via a coupling parameter β ą 0 (the inverse temperature). The model is defined through a Gibbs measure,

dP ω n,β dP psq :" 1 Z ω n β exp ´β n ÿ i"1 ω i,s i ¯, (1.1)
where Z ω n β is the partition function of the model. One of the main question about this model is that of the localization and super-diffusivity of paths trajectories drawn from the measure P ω n,β . The transversal exponent ξ describes the fluctuation of the end-point, that is EE ω n,β |S n | « n ξ as n Ñ 8. Another quantity of interest is the fluctuation exponent χ, that describes the fluctuations of log Z ω n,β , i.e. | log Z ω n,β ´E log Z ω n,β | « n χ as n Ñ 8. This model has been widely studied in the physical and mathematical literature (we refer to [START_REF] Comets | Directed Polymers in Random Environments[END_REF][START_REF] Comets | Probabilistic analysis of directed polymers in a random environment: a review. Stochastic analysis on large scale interacting systems[END_REF] for a general overview), in particular when ω n,x have an exponential moment. The case of the dimension d " 1 as attracted much attention in recent years, in particular because the model is in the KPZ universality class (log Z ω n,β is seen as a discretization of the Hopf-Cole solution of the KPZ equation). It is conjectured that the transversal and fluctuation exponents are ξ " 2{3 and χ " 1{3 respectively. Moreover, it is expected that the point-to-point partition function, when properly centered and renormalized, converges in distribution to the GUE distribution. Such scalings has been proved so far only for some special models, cf. [START_REF] Balázs | Fluctuation exponent of the kpz/stochastic burgers equation[END_REF][START_REF] Seppäläinen | Scaling for a one-dimensional directed polymer with boundary conditions[END_REF].

A recent and fruitful approach to proving universality results for this model has been to consider is weak-coupling limit, that is when the coupling parameter β is close to criticality. This means that we allow β " β n to depend on n, with β n Ñ 0 as n Ñ 8. In [START_REF] Alberts | The continuum directed random polymer[END_REF][START_REF] Alberts | The intermediate disorder regime for directed polymers in dimension 1 + 1[END_REF] and [START_REF] Caravenna | Polynomial chaos and scaling limits of disordered systems[END_REF], the authors let β n " p βn ´γ , γ " 1{4 for some fixed p β ą 0, and they prove that the model (one may focus on its partition function Z ω n,βn ) converges to a non-trivial (i.e. disordered) continuous version of the model. This is called the intermediate disorder regime, since it somehow interpolates between weak disorder and strong disorder behaviors. More precisely, they showed that log Z ω n,βn ´nλpβ n q pdq ÝÑ log Z ?

2 p β , as n Ñ 8,
where λpsq :" log Ere sω s. The process p β Þ Ñ log Z ?

2 p
β is the so called cross-over process, and interpolates between Gaussian and GUE scalings as p β goes from 0 to 8 (see [START_REF] Amir | Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions[END_REF]). These results were obtained under the assumption that ω has exponential moments, but the universality of the limit was conjectured to hold under the assumption of six moments [START_REF] Alberts | The intermediate disorder regime for directed polymers in dimension 1 + 1[END_REF]. In [START_REF] Dey | High temperature limits for p1 `1q-dimensional directed polymer with heavytailed disorder[END_REF] Dey and Zygouras proved this conjecture, and they suggest that this result is a part of a bigger picture (when λpsq is not defined a different centering is necessary).

1.2. The case of a heavy-tail environment. In the rest of the paper we will focus on the dimension d " 1 for simplicity. We consider the case where the environment distribution ω is non-negative (for simplicity, nothing deep is hidden in that assumption) and has some heavy tail distribution: there is some α ą 0 and some slowly varying function Lp¨q such that P pω ą xq " Lpxqx ´α .

(1.2)

In the case where β ą 0 does not depend on n, the ξ " 2{3, χ " 1{3 picture is expected to be modified, depending on the value of α. According to the heuristics (and terminology) of [START_REF] Biroli | Extreme value problems in matrix theory and other disordered systems[END_REF][START_REF] Geudré | Ground-state statistics of directed polymers with heavy-tailed disorder[END_REF], three regimes should occur, with different paths behaviors: (a) if α ą 5, there should be a collective optimization and we should have ξ " 2{3, KPZ universality class, as in the finite exponential moment case;

(b) if α P p2, 5q, the optmization strategy should be elitist: most of the total energy collected should be via a small fraction of the points visited by the path, and we should have ξ " α`1 2α´1 ; (c) if α P p0, 2q, the strategy is individual: the polymer targets few exceptional points, and we have ξ " 1. This case is treated in [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF][START_REF] Hambly | Heavy tails in last-passage percolation[END_REF].

As suggested by [START_REF] Dey | High temperature limits for p1 `1q-dimensional directed polymer with heavytailed disorder[END_REF], this is part of a larger picture, when the inverse temperature β is allowed to depend on n. Setting β n " p βn ´γ for some p β ą 0 and some γ P R then we have three different classes of coupling. When γ " 0 we recover the standard directed polymer model, when γ ą 0 we have a weak-coupling limit, while in the case γ ă 0 we have a strong-coupling limit. Let us stress that this last case has not been studied in the literature (for no apparent reason) and should also be of interest. In [START_REF] K. Khanin | Intermediate disorder regime for directed polymers in dimension 1+1[END_REF] and in [START_REF] Dey | High temperature limits for p1 `1q-dimensional directed polymer with heavytailed disorder[END_REF], the authors suggest that the fluctuation exponent depends on α, γ in the following manner

ξ " $ & % 2p1´γq 3
for α ě 5´2γ 1´γ , ´1 2 ď γ ď 1 4 ,

1`αp1´γq 2α´1

for α ď 5´2γ 1´γ , 2 α ´1 ď γ ď 3 2α .

(1.

3)

The first part is derived in [START_REF] K. Khanin | Intermediate disorder regime for directed polymers in dimension 1+1[END_REF], based on Airy process considerations, and the second part is derived in [START_REF] Dey | High temperature limits for p1 `1q-dimensional directed polymer with heavytailed disorder[END_REF], based on a Flory argument inspired by [START_REF] Biroli | Extreme value problems in matrix theory and other disordered systems[END_REF]. Moreover, in the two regions of the pα, γq plane defined by (1.3), the KPZ scaling relation χ " 2ξ ´1 should hold (this has been proved in the case γ " 0, α ą 2 in [START_REF] Auffinger | The scaling relation χ " 2ξ ´1 for directed polymers in a random environment[END_REF]). Outside of these regions, one should have ξ " 1{2 (γ large) or ξ " 1 (γ small). This is summarized in Figure 1 below, which is the analogous of [13, Fig. 1]. FIGURE 1. We identify four regions in the pα, γq plane. Region A with α ă 2 is treated in [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF] and Region B with α ą 1{2 in [START_REF] Dey | High temperature limits for p1 `1q-dimensional directed polymer with heavytailed disorder[END_REF]. Regions C and D are still open, and the KPZ scaling relation χ " 2ξ ´1 should hold in these two regions. Our main result is to settle the picture in the case α P p0, 2q.

This picture is far from being settled, and so far only the border cases where ξ " 1 or ξ " 1{2 have been studied: Dey and Zygouras [START_REF] Dey | High temperature limits for p1 `1q-dimensional directed polymer with heavytailed disorder[END_REF] proved that ξ " 1{2 in the cases α ą 6, γ " 1{4 and α P p1{2, 6q, γ " 3{2α; Auffinger and Louidor [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF] proved that ξ " 1 for α P p0, 2q and γ " 2 α ´1. Here, we complete the picture in the case α P p0, 2q. For α P p1{2, 2q we go beyond the cases ξ " 1{2 or ξ " 1: we identify the correct order for the transversal fluctuations (they interpolate between ξ " 1{2 and ξ " 1), and we prove the convergence of log Z ω n,βn in all possible intermediate disorder regimes-this proves Conjecture 1.7 in [START_REF] Dey | High temperature limits for p1 `1q-dimensional directed polymer with heavytailed disorder[END_REF]. For α ă 1{2 we show that a sharp transition occurs on the line γ " 2 α ´1, between a regime where ξ " 1 and a regime where ξ " 1{2.

MAIN RESULTS: WEAK-COUPLING LIMITS IN THE CASE α P p0, 2q

From now on, we consider the case of an environment ω verifying (1.2) with α P p0, 2q. For the inverse temperature, we will consider arbitrary sequences pβ n q ně1 , but a reference example is β n " n ´γ for some γ P R.

For two sequences pa n q ně1 , pb n q ně1 , we use the notations (2.1)

a n " b n if lim nÑ8 a n {b n " 1, a n ! b n if lim nÑ8 a n {b n " 0, and a n -b n if 0 ă lim inf a n {b n ď lim sup a n {b n ă 8.
Note that the second identity characterizes mpxq up to asymptotic equivalence: we have that mp¨q is a regularly varying function with exponent 1{α.

Assuming that the transversal fluctuations are of order h n (we necessarily have ? n ď h n ď n), then the amount of weight collected by a path should be of order mpnh n q (it should be dominated by the maximal value of ω in r0, ns ˆr´h n , h n s). On the other hand, thanks to moderate deviations estimates for the simple random walk, the entropic cost of having fluctuations of order h n is roughly h 2 n {n at the exponential level -at least when h n " ? n log n, see (2.14) below. It therefore leads us to define h n (seen as a function of β n ) up to asymptotic equivalence by the relation

β n mpnh n q " h 2 n {n . (2.2)
In the case β n " n ´γ and α P p1{2, 2q we recover (1.3), that is we get that h n " n ξ`op1q with ξ " 1`αp1´γq 2α´1 , which is in p1{2, 1q for γ P p 2 α ´1, 3 2α q. When α P p0, 1{2q, there is no h n verifying (2.2) with ? n ! h n ! n, leading to believe that intermediate transversal fluctuations (i.e. ξ P p1{2, 1q) cannot occur. In the following, we separate the cases α P p1{2, 2q and α P p0, 1{2q.

2.2.

A natural candidate for the scaling limit. Once we have identified in (2.2) the scale h n for the transversal fluctuations, we are able to rescale both path trajectories and the field pω i,x q, so that we can define the rescaled "entropy" and "energy" of a path, and the corresponding continous quantities. The rescaled paths will be in the following set

D :" s : r0, 1s Ñ R ; s continuous and a.e. differentiable ( , (2.3)
and the (continuum) entropy of a path s P D will derive from the rate function of the moderate deviation of the simple random walk (see [START_REF] Stone | On local and ratio limit theorems[END_REF] or (2.14) below), i.e.

Entpsq " 1 2

ż 1 0 `s1 ptq ˘2dt for s P D. (2.4)
As far as the disorder field is concerned, we let P :" tpw i , t i , x i qu iě1 be a Poisson Point Process on r0, 8q ˆr0, 1s ˆR of intensity µpdwdtdxq " α 2 w ´α´1 1 twą0u dwdtdx. For a quenched realization of P, the energy of a continuous path s P D is then defined by πpsq " π P psq :"

ÿ pw,t,xqPP w 1 tpt,xqPsu , (2.5)
where the notation pt, xq P s means that s t " x.

Then, a natural guess for the continuous scaling limit of the partition function is to consider an energy-entropy competition variational problem. For any β ě 0 we let

T β :" sup sPD,Entpsqă`8 ! βπpsq ´Entpsq ) . (2.6)
This variational problem was originally introduced by Dey and Zygouras [13, Conjecture 1.7], conjecturing that it was well defined as long as α P p1{2, 2q and that it was the good candidate for the scaling limit. In [ Let us mention here that in [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF], the authors consider the case of transversal fluctuations of order n. The natural candidate for the limit is p T β , defined analogously to (2.6) by p T β " 0 for β " 0, and for β ą 0

p T β " sup sPLip 1 ! πpsq ´1 β p Entpsq ) . (2.7)
Here the supremum is taken over the set Lip e `s1 ptq ˘dt with epxq " 1 2 p1 `xq logp1 `xq `1 2 p1 ´xq logp1 ´xq .

2.3.

Main results I : the case α P p1{2, 2q. Our first result deals with the transversal fluctuations of the polymer: we prove that h n defined in (2.2) indeed gives the correct order for the transversal fluctuations.

Theorem 2.2. Assume that α P p1{2, 2q, that β n mpn 2 q Ñ 0 and that β n mpn 3{2 q Ñ `8, and define h n as in (2.2): then ? n ! h n ! n. Then, there are constants c 1 , c 2 and ν ą 0 such that for any sequences A n ě 1 we have for all n ě 1

P ˆPω n,βn `max iďn |S i | ě A n h n ˘ě n e ´c1 A 2 n h 2 n {n ˙ď c 2 A ´ν n . (2.8)
In particular, this proves that if h n defined in (2.2) is larger than a constant times ? n log n, then ne ´c1 Ah 2 n {n goes to 0 as n Ñ 8 provided that A is large enough: the transversal fluctuations are at most Ah n , with high P-probability. On the other hand, if h n is much smaller than ? n log n, then this theorem does not give sharp information: we still find that the transversal fluctuations must be smaller than A ? n log n, with high P-probability. Anyway, in the course of the demonstration of our results, it will be clear that the main contribution to the partition function comes from trajectories with transversal fluctuations of order exactly h n .

We stress that the cases β n mpn 2 q Ñ β P p0, `8s and β n mpn 3{2 q Ñ β P r0, 8q have already been considered by Auffinger and Louidor [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF] and Dey and Zygouras [START_REF] Dey | High temperature limits for p1 `1q-dimensional directed polymer with heavytailed disorder[END_REF] respectively: they find that the transversal fluctuations are of order n, resp.

? n. with T 1 defined in (2.6).

We stress here that we need to recenter log Z ω n,βn by nβ n Erωs only when necessary, that is when n{mpnh n q does not go to 0: in terms of the picture described in Figure 1, this can happen only when γ ě 4 ´2α, and in particular when α ě 3{2 (this is stressed in the statement of the theorem). (2.11)

When r " 0 we denote by r T β the quantity r T pě0q β

. In Proposition 5.5 below, we prove that these quantities are well defined, and that there exists β c " β c pPq P p0, 8q such that

r T β P p0, 8q if β ą β c and r T β " 0 if β ă β c . Theorem
(Recall that β n mpnh n q " h 2 n {n " β log n.) If r
T β ą 0 (β ą β c ) the scaling limit is therefore well identified, and log Z ω n,βn (when recentered) grows like β r T β log n with β r T β ą 0. On the other hand, if r T β " 0, then the above theorem gives only a trivial limit. By an extended version of Skorokhod representation theorem [START_REF] Kallenberg | Foundations of Modern Probability[END_REF]Corollary 5.12], one can couple the discrete environment and the continuum field P in order to obtain an almost sure convergence in Theorem 2.5 above. Hence, it makes sense to work conditionally on r T ě1 as n Ñ 8 .

β ă 0 (β ă β c ),
Here, W pαq β is some specific α-stable random variable (defined in [13, p. 4011]). Some comments about the different regimes. The regimes 2-3-4 have different behavior due to the different behaviors for the local moderate deviation, see [START_REF] Stone | On local and ratio limit theorems[END_REF]Theorem 3]. We indeed have that for

? n ! h n ! n p n ph n q :" PpS n " h n q " c ? n exp ´´p1 `op1qq h 2 n 2n ¯, (2.14)
so that we identify three main possibilities: if h n ! ? n log n, then p n ph n q " n ´1{2`op1q ; if h n " c ? n log n then p n ph n q " n ´pc 2 `1q{2`op1q ; if h n " ? n log n then p n ph n q " e ´p1`op1qqh 2 n {n which decays faster than any power of n. This is actually reflected in the behavior of the partition function. Let us denote Zω n,βn " e ´nβnCα ˆZω n,βn be the renormalized (when necessary) partition function. We recall that C α is equal either to Erωs1 tαě3{2u (Regime 2 and 3-a) or to Erω1 tωď1{βnu s1 αě1 (Regime 3-b and 4). Then we have going to infinity slower than any power of n: this corresponds to the cost for a trajectory to visit a single site, at which the supremum in W 1 is attained. In Regime 5, Zω n,βn goes to 1 with a correction of order n ´1{2 . 2.4. Main results II : the case α P p0, 1{2q. In this case, since we have n ´1mpn 2 q{mpn 3{2 q Ñ 8, there is no sequence β n such that β n n ´1mpn 2 q Ñ 0 and β n mpn 3{2 q Ñ `8. First of all, Theorem 2.3 already gives a result, but a phase transition has been identified in [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF][START_REF] Torri | Pinning model with heavy tailed disorder[END_REF] when α P p0, 1{2q. Theorem 2.9 ([5, 23]). When α P p0, 1{2q, p T β defined in (2.7) undergoes a phase transition: there exists some p β c " p β c pPq with p β c P p0, 8q P-a.s., such that p

T β " 0 if β ď p β c and p T β ą 0 if β ą p β c .
The fact that p T p βc " 0 was not noted in [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF][START_REF] Torri | Pinning model with heavy tailed disorder[END_REF], but simply comes from the (left) continuity of β Þ Ñ p T β (the proof is identical to that for β Þ Ñ T β , see [START_REF] Berger | Entropy-controlled last-passage percolation[END_REF]Section 4.5]). In view of Theorem 2. is well defined and has an α-stable distribution, with explicit characteristic function, see Lemma 1.3 in [START_REF] Dey | High temperature limits for p1 `1q-dimensional directed polymer with heavytailed disorder[END_REF]. Theorem 2.10 therefore shows that, when α ă 1{2, a very sharp phase transition occurs on the line β n " βn{mpn 2 q: for β ď p β c , transversal fluctuations are of order ? n whereas for β ą p β c they are of order n.

2.5. Some comments and perspectives. We now present some possible generalizations, and we discuss some open questions.

About the case α " 1{2. We excluded above the case α " 1{2. In that case, both n ´1mpn 2 q and mpn 3{2 q are regularly varying with index 3, and there are mostly two possibilities.

(1) If n ´1mpn 2 q mpn 3{2 q Ñ 0 (for instance if Lpxq " e ´plog xq b for some b P p0, 1q), there are sequences pβ n q ně1 with β n n ´1mpn 2 q Ñ 0 and β n mpn 3{2 q Ñ `8. The situation should be similar to that of Section 2.3: there should be five regimes, with transversal fluctuations h n interpolating between

? n and n.

(2) If n ´1mpn 2 q mpn 3{2 q Ñ c P p0, 8s (for instance if Lpnq " plog xq b for some b), there is no sequence pβ n q ně1 with β n n ´1mpn 2 q Ñ 0 and β n mpn 3{2 q Ñ `8. Then, the situation should be similar to that of Section 2.4: there should be only two regimes, with transversal fluctuations either

? n or n.

Toward the case α P p2, 5q. When α P p2, 5q (more generally in region C in Figure 1), an important difficulty is to find the correct centering term for log Z ω n,βn . Another problem is that the variational problem T β defined in (2.6) is T β " `8 a.s., since paths that collect many small weights bring an important contribution to T β . The main objective is therefore to prove a result of the type: there exists a function f p¨q such that, for α P p2, 6q and any β n in region C of Figure 1 1

β n mpnh n q ´log Z ω n,βn ´f pβ n q ¯pdq ÝÑ Ť1 ,
with h n defined as in (2.2) and where Ť1 is somehow a "recentered" version of the variational problem (2.6) (that is in which the contribution of the small weights has been canceled out). The difficulties are however serious: one needs (i) to identify the centering term f pβ n q, (ii) to make sense of the variational problem Ť1 .

Path localization. We mention that in [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF], Auffinger and Louidor show some path localization: they prove that, under P ω n,βn , path trajectories concentrate around the (unique) maximizer γ n,βn of the discrete analogue of the variational problem (2.7), see Theorem 2.1 in [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF]; moreover this maximizer γ n,βn converges in distribution to the (unique) maximizer p γ β of the variational problem (2.7). This could theoretically be done in our setting: in [7, Section 4.6] we prove the existence and uniqueness of the maximizer of the continuous variational problem (2.6). Then similar techniques to those of [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF] could potentially be used, and one would obtain a result analogous to [5, Thm. 2.1] Higher dimensions. Similarly to [START_REF] Auffinger | Directed polymers in random environment with heavy tails[END_REF], our methods should work in any dimension 1 `d (one temporal dimension, d transversal dimensions). The relation (2.2) is replaced by β n mpnh d n q " h 2 n {n: for paths with transversal scale h n , the energy collected should be of order β n mpnh d n q while the entropy cost should remain of order h 2 n {n, at the exponential level. For α P p0, 1 `dq, and choosing β n " n ´γ , we should therefore find that in dimension d a similar picture to Figure 1 hold:

Case α P p0, d{2q γ ă 1`d α ´1 γ ą 1`d α ´1 ξ " 1 ξ " 1{2 Case α P pd{2, 1 `dq γ ď 1`d α ´1 1`d α ´1 ă γ ă 2`d 2α γ ě 2`d 2α ξ " 1 ξ " 1`p1´γqα 2α´d P p 1 2 , 1q ξ " 1 2
2.6. Organization of the rest of the paper. We present an overview of the main ideas used in the paper, and describe how the proofs are organized.

˚In Section 3, we recall some of the notations and results of the Entropy-controlled Last-Passage Percolation (E-LPP) developed in [START_REF] Berger | Entropy-controlled last-passage percolation[END_REF], which will be a central tool for the rest of the paper. In particular, we introduce a discrete energy/entropy variational problem (3.3) (which is the discrete counterpart of (2.6)), and state its convergence toward (2.6) in Proposition 3.1.

˚In Section 4, we prove Theorem 2.2, identifying the correct transversal fluctuations. In order to make our ideas appear clearer, we first treat the case when no centering is needed (i.e. α ă 3{2) in Section 4.1. In Section 4.2 we adapt the proof to the case where it is needed. In the first case, we use a rough bound

P ω n,βn `max iďn |S i | ě A n h n ˘ď Z ω n,βn `max iďn |S i | ě A n h n ˘, the second term being the partition function restricted to trajectories with max iďn |S i | ě A n h n .
The key idea is to decompose this quantity into subparts where trajectories have a "fixed" transversal fluctuation

Z ω n,βn `max iďn |S i | ě A n h n ˘" log 2 pn{hnq ÿ k"log 2 An`1 Z ω n,βn ´max iďn |S i | P r2 k´1 h n , 2 k h n q ¯.
Then, we control each term separately. Forcing the random walk to reach the scale 2 k´1 h n has an entropy cost expp´c2 2k h 2 n {nq so we need to understand if the partition function, when restricted to trajectories with max iďn |S i | ď 2 k h n , compensates this cost (cf. (4.3)): we need to estimate the probability of having Z ω n,βn pmax iďn |S i | ď 2 k h n q ě e c2 2k h 2 n {n . This is the purpose of Lemma 4.1, which is the central estimate of this section, and which tediously uses estimates derived in [START_REF] Berger | Entropy-controlled last-passage percolation[END_REF] (in particular Proposition 2.6).

˚In Section 5, we consider Regimes 2 and 3-a, and we prove Theorems 2.4-2.5. The proof is decomposed into three steps. In the first step (Section 5.1), we use Theorem 2.2 in order to restrict the partition function to path trajectories that have transversal fluctuations smaller than Ah n (for some large A fixed). In a second step (Section 5.2), we show that we can keep only the largest weights in the box of height Ah n (more precisely a finite number of them), the small-weights contribution being negligible. Finally, the third step (Section 5.3) consists in proving the convergence of the large-weights partition function, and relies on the convergence of the discrete variational problem of Section 3.

˚In Section 6, we treat Regime 3-b and Regime 4, and we prove Theorems 2.6-2.7. We proceed in four steps. In the first step (Section 6.1), we again use Theorem 2.2 to restrict the partition function to trajectories with transversal fluctuations smaller than A ? n log n (for some large A fixed). The second step (Section 6.2) consists in showing that one can restrict to large weights. In the third step (Section 6.3), we observe that since we consider a regime log Z ω n,βn Ñ 0, it is equivalent to studying the convergence of Z ω n,βn ´1: we reduce to showing the convergence of a finite number of terms of the polynomial chaos expansion of Z ω n,βn ´1, see Lemmas 6.2-6.3. We prove this convergence in a last step: in Section 6.4, we show the convergence in Regime 3-b (Lemma 6.2), relying on the convergence of a discrete variational problem. In Section 6.5, we show the convergence in Regime 4 (Lemma 6.3), which is slightly more technical since we first need to reduce to trajectories with transversal fluctuations of order h n ! ? n log n.

˚In Section 7, we consider the case α P p0, 1{2q, and we prove Theorem 2.10. First, in Section 7.1, we prove (2.15) i.e. that there cannot be intermediate transversal fluctuations between ? n and n. We use mostly the same ideas as in Section 4, decomposing the contribution to the partition function according to the scale of the path, and controlling the entropic cost vs. energy reward for each term. Here, some simplifications occur: one can bound the maximal energy collected by a path at a given scale by the sum of all weights in a box containing the path, this sum being roughly dominated by the maximal weight in the box (this is true for α ă 1). We then turn to the convergence of the partition function in Section 7.2. The idea is similar to that of [13, Section 5], and consists in several steps: first we reduce the partition function to trajectories that stay at scale ? n log n; then we perform a polynomial chaos expansion of Z ω n,βn ´1 and we show that only the first term contributes; finally, we prove the convergence of the main term, see Lemma 7.2, showing in particular that the main contribution comes from trajectories that stay at scale ? n.

DISCRETE ENERGY-ENTROPY VARIATIONAL PROBLEM

We introduce here a few necessary notations, and state some useful results from [START_REF] Berger | Entropy-controlled last-passage percolation[END_REF]. Let us consider a box Λ n,h " 1, n ˆ ´h, h . For any set ∆ Ă Λ n,h , we define the (discrete) energy collected by ∆ by Ω n,h p∆q :"

ÿ pi,xqP∆ ω i,x . (3.1)
We can also define the (discrete) entropy of a finite set ∆ " pt i , x i q; 1 ď i ď j ( Ă R 2 with |∆| " j P N and with 0 ď t 1 ď t 2 ď ¨¨¨ď t j (with t 0 " 0, x 0 " 0) Entp∆q :" 1 2

j ÿ i"1 px i ´xi´1 q 2 t i ´ti´1 , (3.2
T β n,h n,h :" max ∆ĂΛ n,h β n,h Ω n,h p∆q ´Entp∆q ( , (3.3)
with β n,h some function of n, h (soon to be specified).

We may rewrite the disorder in the region Λ n,h , using the ordered statistic: we let M pn,hq r be the r-th largest value of pω i,x q pi,xqPΛ n,h and Y pn,hq r P Λ n,h its position. In such a way pω i,j q pi,jqPΛn "pM pn,hq r , Y pn,hq

r q |Λ n,h | r"1 . (3.4)
In the following we refer to pM pn,hq r q |Λ n,h | r"1 as the weight sequence. Note also that pY

pn,hq r q |Λ n,h | r"1
is simply a random permutation of the points of Λ n,h . The ordered statistics allows us to redefine the energy collected by a set ∆ Ă Λ n,h , and its contribution by the first weights (with

1 ď ď |Λ n,h |) by Ω p q n,h p∆q :" ÿ r"1 M pn,hq r 1 tY pn,hq r P∆u , Ω n,h p∆q :" Ω p|Λ n,h |q n,h p∆q . (3.5)
We also set Ω pą q n,h p∆q :" Ω n,h p∆q ´Ωp q n,h p∆q. We then define analogues of (3.3) with a restriction to the largest weights, or beyond the -th weight

T β n,h ,p q n,h :" max ∆ĂΛ n,h β n,h Ω p q n,h p∆q ´Entp∆q ( , T β n,h ,pą q n,h :" max ∆ĂΛ n,h β n,h Ω pą q n,h p∆q ´Entp∆q ( . (3.6)
Estimates on these quantities are given in [7, Prop. with M q :" ts P D, Entpsq ă 8, max tPr0,1s |sptq| ď qu. We also have

n h 2 T
β n,h ,p q n,qh pdq ÝÑ T p q ν,q :" sup sPMq νπ p q psq ´Entpsq ( as n Ñ 8,

where π p q :" ř r"1 M r 1 tYrPsu with tpM r , Y r qu rě1 the ordered statistics of P restricted to r0, 1s ˆr´q, qs, see [7, Section 5.1] for details.

Finally, we have T p q ν,q Ñ T ν,q as Ñ 8, and T ν,q Ñ T ν as q Ñ 8, a.s.

TRANSVERSAL FLUCTUATIONS: PROOF OF THEOREM 2.2

In this section, we have α P p1{2, 2q. First, we partition the interval rA n h n , ns into blocks

B k,n :" r2 k´1 h n , 2 k h n q, k " log 2 A n `1, . . . , log 2 pn{h n q `1. (4.1)
In such a way,

P ω n,βn `max iďn |S i | ě A n h n ˘" log 2 pn{hnq ÿ k"log 2 An`1 P ω n,βn `max iďn |S i | P B k,n ˘. (4.2)
We first deal with the case where n{mpnh n q nÑ8 Ñ 0 for the sake of clarity of the exposition: in that case, log Z ω n,βn does not need to be recentered. We treat the remaining case (in particular we have α ě 3{2) in a second step. 4.1. Case n{mpnh n q nÑ8 Ñ 0. We observe that the assumption ω ě 0 implies that the partition function Z ω n,βn is larger than one. Therefore,

P ω n,βn `max iďn |S i | P B k,n ˘ď Z ω n,βn `max iďn |S i | P B k,n ˘.
By using Cauchy-Schwarz inequality, we get that

Z ω n,βn `max iďn |S i | P B k,n ˘2 ď P `max iďn ˇˇS i ˇˇě 2 k´1 h n ˘ˆZ ω n,2βn `max iďn |S i | ď 2 k h n ˘. (4.3)
The first probability is bounded by 2Pp|S n | ě h n q ď 4 expp´2 2k h 2 n {2nq (by Levy's inequality and a standard Chernov's bound). We are going to show the following lemma, which is the central estimate of the proof.

Lemma 4.1.

There exist some constant q 0 ą 0 and some ν ą 0, such that for all q ě q 0 we have

P ´Zω n,2βn `max iďn |S i | ď qh n ˘ě e 1 4 q 2 h 2 n n ¯ď q ´ν ´1 `1 ^n mpnh n q ¯. (4.4)
Therefore, if n{mpnh n q nÑ8 Ñ 0, this lemma gives that for c 0 " 1{8 and for k large enough (i.e. A n large enough), using (4.3),

P ´Zω

n,βn `max

iďn |S i | P B k,n ˘ě 4e ´c0 2 2k h 2 n {n ¯ď p2 k q ´ν .
Then, using that ř kąlog 2 An 4e ´c0 2 2k h 2 n {n ď e ´c1 A 2 n h 2 n {n , we get that by a union bound

P ´Pω n,β `max iďn ˇˇS i ˇˇě A n h n ˘ě e ´c1 A 2 n h 2 n {n ď log 2 pn{hnq ÿ k"log 2 An`1 P ´Zω n,β `max iďn ˇˇS i ˇˇP B k,n ˘ě 4e ´c0 2 2k h 2 n {n ď ÿ kąlog 2 An 2 ´νk ď cA ´ν n . (4.5)
We stress that in the case when n{mpnh n q nÑ8 Ñ 0, we do not need the additional n in front of e ´c1 A 2 n h 2 n {n in (2.8).

Proof of Lemma 4.1. For simplicity, we assume in the following that qh n is an integer. We fix δ ą 0 such that p1 `δq{α ă 2 and p1 ´δq{α ą 1{2, and let

T " T n pqh n q " h 2 n n q 1{α pq 2 h 2 n {nq ´p1´δq 3{2 {α _ 1 (4.6)
be a truncation level. Note that if α ď p1 ´δq 3{2 then we have T " 1. We decompose the partition function into three parts: thanks to Hölder's inequality, we can write that

log Z ω n,2βn `max iďn |S i | ď qh n ˘ď 1 3 log Z pąTq n,6βn `1 3 log Z pp1,Tsq n,6βn `1 3 log Z pď1q n,6βn , (4.7)
where the three partition functions correspond to three ranges for the weights β n ω i,S i :

Z pąTq n,6βn :" E " exp ´n ÿ i"1 6β n ω i,S i 1 tβnω i,S i ąTu ¯1tmax iďn |S i |ďqhnu ı (4.8) Z pp1,Tsq n,6βn :" E " exp ´n ÿ i"1 6β n ω i,S i 1 tβnω i,S i Pp1,Tsu ¯1tmax iďn |S i |ďqhnu ı (4.9) Z pď1q n,6βn :" E " exp ´n ÿ i"1 6β n ω i,S i 1 tβnω i,S i ď1u ¯1tmax iďn |S i |ďqhnu ı . (4.10)
We now show that with high probability, these three partition functions cannot be large. Note that when T " 1, the second term is equal to 1 and we do not have to deal with it.

Term 1. For (4.8), we prove that for any ν ă 2α ´1, for q sufficiently large, for all n large enough we have

P ´log Z pąTq n,6βn ě c 0 q 2 h 2 n n ¯ď q ´ν . (4.11)
We compare this truncated partition function with the partition function where we keep the first weights in the ordered statistics pM pn,qhnq i q 1ďiďnqhn . Define

" n pqh n q :" pq 2 h 2 n {nq 1´δ , so T "

h 2 n n q 1{α ˆ ´p1´δq 1{2 {α , (4.12)
and set

Z p q n,6βn :" E " exp
´ ÿ

i"1

6β n M pn,qhnq i 1 tY pn,qhnq i PSu ¯ı . (4.13)
Remark that, with the definition of T and thanks to the relation (2.2) verified by β n , we have that for n large enough P ´βn M pn,qhnq ą T ¯ď P ´M pn,qhnq ě 1 2 q 1{α ´p1´δq 1{2 {α mpnh n q

Then, since we have q{ ď 1 (see (4.12)), we can use Potter's bound to get that for n sufficiently large m `nqh n { ˘ď pq{ q p1´δ 2 q{α mpnh n q , and we obtain that provided that δ is small enough

P ´βn M pn,qhnq ą T ¯ď P ´M pn,qhnq ě c 0 q δ 2 {α δ 2 {α m `nqh n { ˘¯ď pcq q ´δ2 {2 ,
where we used [7, Lemma 5.1] for the last inequality. We therefore get that, with probability larger than 1 ´pc q ´δ {2 (note that ´δ {2 ď q ´δ {2 ď q ´4 for n large enough), we have that

! pi, xq P 1, n ˆ ´qh n , qh n ; β n ω i,x ą T ) Ă Υ :" Y pn,qhnq 1 , . . . , Y pn,qhnq ( , (4.14)
and hence Z pąT q n,6βn ď Z p q n,6βn . We are therefore left to focus on the term Z p q n,6βn : recalling the definitions (3.5) and (3.6), we get that

Z p q n,6βn " ÿ ∆ĂΥ e 6βnΩ p q n,qhn p∆q P `S X Υ m " ∆ ď ÿ ∆ĂΥ exp `6β n Ω n,qhn p∆q ´Entp∆q ˘ď 2 exp ´T 6βn,p q n,qhn ¯, (4.15)
where we used that Pp∆ Ă Sq ď expp´Entp∆qq as noted below (3.2). Note that we have ď 1 2 c 0 q 2 h 2 n {n for n large enough (and q ě 1), so we get that

P ´log Z p q n,6βn ě c 0 q 2 h 2 n n ¯ď P ´T 6βn,p q n,qhn ě 1 2 c 0 q 2 h 2 n n ¯.
Then, by the definition (2.2) and thanks to Potter's bound, for any η ą 0 there exists a constant c η such that for any q ě 1

`6β n mpnqh n q ˘4{3 pq 2 h 2 n {nq 1{3 ď c η q p1`ηq 4 3α ´2 3 h 2 n n " c η pq 4{3 q p1`ηq{α´2 ˆq2 h 2 n n ,
where we used that for any η ą 0, mpnqh n q ď c 1 η q p1`ηq{α mpnh n q provided that n is large enough (Potter's bound). Therefore, provided that η is small enough so that p1 ὴq{α ă 2, an application of [7, Prop. 2.6] gives that for q large enough (so that b q :" c 0 2cη pq 4{3 q 2´p1`ηq{α is large),

P

´T 6βn,p q n,qhn ě

1 2 c 0 q 2 h 2 n n ¯ď P ´T 6βn,p q n,qhn ě b q ˆ`6β n mpnqh n q ˘4{3 pq 2 h 2 n {nq 1{3 ¯ď cq ´ν , (4.16)
with ν " 2α ´1 ´2η. This gives (4.11), since η is arbitrary.

Term 2. We now turn to (4.9) We consider only the case T ą 1 (and in particular we have α ą p1 ´δq 3{2 ). We show that for any η ą 0, there is a constant c η ą 0 such that for q large enough and n large enough,

P ´log Z pp1,Tsq n,6βn ě c 0 `q2 h 2 n {n ˘1´η ¯ď exp `´c η pq 2 h 2 n {nq 1{3 ˘. (4.17)
Again, we need to decompose Z pp1,Tsq n,6βn according to the values of the weights. We set θ :" p1 ´δq2{α ą 1, and let j :" pq 2 h 2 n {nq θ j p1´δq " p 0 q θ j , with 0 " " pq 2 h 2 n {nq 1´δ as in (4.12) (4.18)

T pjq :" h 2 n n q 1{α ˆpq 2 h 2 n {nq ´θj p1´δq 3{2 {α " h 2 n n q 1{α ` j ˘´p1´δq 1{2 {α (4.19)
for j P t0, . . . , κu with κ the first integer such that θ κ ą α{p1 ´δq 3{2 . We get that T p0q " T, and T pκq ă 1. Then, thanks to Hölder inequality we may write

log Z pp1,T sq n,6βn ď 1 κ κ ÿ j"1 log Z ppT pjq ,T pj´1q sq n,6κβn
, with Z ppT pjq ,T pj´1q sq n,6κβn

:" E " exp ´n ÿ

i"1

6κβ n ω i,S i 1 tβnω i,S i PpT pjq ,T pj´1q su ¯1tmax iďn |S i |ďqhnu ı .
To prove (4.17), it is therefore enough to prove that for any 1 ď j ď κ, since j ě pq 2 h 2 n {nq 1´δ ,

P ´log Z ppT pjq ,T pj´1q sq n,6κβn ě 8κ `q2 h 2 n {n ˘ ´δ{10 j ¯ď exp `´cpq 2 h 2 n {nq 1{3 ˘. (4.20)
First of all, we notice that in view of (4. On this event, and using that j " p j´1 q p1´δq2{α and

T pj´1q " h 2 n n q 1{α ´p1´δq ´1{2 {2 j ď h 2 n n q 1{α ´1{2´δ{5 j (if δ is small), we have Z ppT pjq ,T pj´1q sq n,6κβn ď E " exp ´6κT pj´1q j ÿ i"1 1 tY pn,qhnq i PSu ¯ı (4.22) ď e 6κq 2 h 2 n n ´δ{10 j `Hj with
H j :" j ÿ k"q 2´1 α 1{2`δ{10 j ÿ ∆ĂΥ j ;|∆|"k e 6κ h 2 n n q 1{α ´1{2´δ{5 j k P `S X Υ j " ∆ ď j ÿ k"q 2´1 α 1{2`δ{10 j ˆ j k ˙exp ´6κ h 2 n n q 1{α ´1{2´δ{5 j k ´inf ∆ĂΥ j ,|∆|"k Entp∆q ¯.
Then, we may bound ` j k ˘ď e k log j . We notice from the definition of κ (and since θ P p1, 2q) that there exists some η ą 0 such that j ď κ ď pq 2 h 2 n {nq 2´η for any 1 ď j ď κ: it shows in particular that log j ď δ 2 j ď q 2 h 2 n n ´1{2´δ{5 j

, provided that n is sufficiently large and δ has been fixed sufficiently small. We end up with the following bound

H j ď j ÿ k"q 2´1 α 1{2`δ{10 j exp ´cq 2 h 2 n n ´1{2´δ{5 j k ´inf ∆ĂΥ j ,|∆|"k Entp∆q ¯.
Then, we may use relation (2.5) of [START_REF] Berger | Entropy-controlled last-passage percolation[END_REF] (with m " j , h " qh n ) to get that, for any k ě q 2´1 α 1{2`δ{10 j P ´inf

∆ĂΥ j ,|∆|"k Entp∆q ď 2cq 2 h 2 n n ´1{2´δ{5 j k ¯ď ˆC0 p2c ´1{2´δ{5 j kq 1{2 j k 2 ˙k ď `cq 3 2α ´3 ´δ{4 j ˘k ď `c j ˘´δk{4 . (4.23)
For the last inequality, we used that q 3 2α ´3 ď 1, since α ą 1{2 and q ě 1. Since we have that q 2 h 2 n n ´1{2´δ{5 j ě 1, we get that there is a constant c 1 ą 0 such that

ÿ kěq 2´1 α 1{2`δ{10 j e ´cq 2 h 2 n n ´1{2´δ{5 j k ď c 1 e ´cq 2 h 2 n n ´δ{10 j ď c 1 .
Using (4.23), we therefore obtain, via a union bound (also recalling (4.21)), that provided that n is large enough

P ´ZppT pjq ,T pj´1q sq n,6κβn ě e 8κq 2 h 2 n n ´δ{10 j ¯ď pc j q ´δ j {4 `ÿ kěq 2´1 α 1{2`δ{10 j `c j ˘´δk{4 ď `c j ˘´c δ 1{2 j .
This proves (4.20) since j ě 0 " pq 2 h 2 n {nq 1´δ . Term 3. For the last part (4.10), we prove that for arbitrary η ą 0,

P ´log Z pď1q n,6βn ě c 0 q 2 h 2 n n ¯ď cq ´2 ˆ# n mpnhnq if α ą 1 , n mpnhnq p1´ηqα if α ď 1 . (4.24)
Let us stress that in the case α ď 1 we get that for n large mpnh n q p1´ηqα ě pnh n q 1´2η , therefore n{pnh n q p1´ηqα goes to 0 provided that η is small enough, since we are considering the case when h n ě ? n. Hence, we can replace the upper bound in (4.24) by 1 ^pn{mpnh n qq. To prove (4.24), we use that e 6x1 txď1u ď 1 `e6 x1 txď1u for any x, and we get that

Z pď1q n,6βn ď E " n ź i"1 `1 `6e 6 β n ω i,s i 1 tβnω i,s i ď1u ˘ı , (4.25)
and

EZ pď1q n,6βn ď E " n ź i"1 `1 `6e 6 β n E " ω1 tωď1{βnu ‰˘ı ď e 6e 6 nβnErω1 tωď1{βnu s .
Therefore, by Markov inequality and Jensen inequality,

P ´log Z pď1q n,6βn ě c 0 q 2 h 2 n n ¯ď 1 c 0 q 2 n h 2 n log EZ pď1q n,6βn ď Cq ´2 n 2 β n h 2 n E " ω1 tωď1{βnu ‰ . ( 4
P ´Z ω n,β `max iďn ˇˇS i ˇˇP B k,n ˘ě 4e ´c0 2 2k h 2 n {n ¯.
Then, we have a lemma which is the analogous of Lemma 4.1 for Zω n,βn .

Lemma 4.3.

There exist some constant q 0 ą 0 and some ν ą 0, such that for all q ě q 0 we have

P ´Z ω n,2βn `max iďn |S i | ď qh n ˘ě e 1 4 q 2 h 2 n n ¯ď q ´ν . (4.30)
Proof. The proof follows the same lines as for Lemma 4.1: (4.7) still holds, with β n ω i,S i replaced by β n pω i,S i ´µq (outside of the indicator function). The bounds (4.11)-(4.17) for terms 1 and 2 still hold, since one fall back to the same estimates by using that pω i,S i ´µq ď ω i,S i . It remains only to control only the third term: we prove that when µ :" Erωs ă 8, then for any δ ą 0, provided that n is large enough,

P ´log Zpď1q n,6βn ě c 0 q 2 h 2 n n ¯ď cq ´2 ˆn´1{2 ´h2 n n ¯α´3 2 `δ , (4.31)
where we set analogously to (4.7)

Zpď1q

n,6βn :" E " exp ´n ÿ

i"1

6β n pω i,S i ´µq1 tβnω i,S i ď1u ¯ı . (4.32)

Then, using h 2 n {n ď n (if α ě 3{2, the upper bound in (4.31) is bounded by cq ´2n α´2`δ which is smaller than q ´2 provided that δ had been fixed small enough.

To prove (4.31), we use that there is a constant c such that e x ď 1 `x `cx 2 as soon as |x| ď 6, so that we get similarly to (

O n :" ! pi, xq P n{2, n ˆ ε 1{2 n h n , 2ε 1{2 n h n ; β n ω i,x ě 2x 2 {i
) .

If the set O n is non-empty, then pick some pi 0 , x 0 q P O n , and consider trajectories which visit this specific site: since all other weights are non-negative (pω´µq1 tβnωą1u ě 0 provided µ ă 1{β n ), we get that We used Stone's local limit theorem [START_REF] Stone | On local and ratio limit theorems[END_REF] for the second inequality (valid provided that n is large, using also that i 0 ě n{2). For the last inequality, we used the definition of O n to bound the argument of the exponential by x 2 0 {i 0 ě ε n h 2 n {n. Therefore, we get that

P ´Z pě1q n,βn ď c ? n e εn h 2 n n ¯ď P `On " H ˘" n ź i"n{2 2ε 1{2 n hn ź x"ε 1{2 n hn ´1 ´P`β n ω ą 2x 2 {i ˘ď ´1 ´P`ω ą 4ε n mpnh n q ˘¯ε 1{2 n nhn
.

For the second inequality we used that x 2 {i ě ε n h 2 n {n for the range considered, together with the relation (2.2) characterizing β n . Then, we use the definition of mpnh n q together with Potter's bound to get that for any fixed δ ą 0, we have P `ω ą 4ε n mpnh n q ˘ě cε ´α`δ n pnh n q ´1, provided that n is large enough. Therefore, we obtain that

P ´Z pě1q n,βn ď c ? n e εn h 2 n n ¯ď exp ´´c ε 1 2 ´α`δ n ¯, (4.39)
which bounds the second term in (4.35).

REGIME 2 AND REGIME 3-A

In this section we prove Theorem 2.4 and Theorem 2.5. We decompose the proof in three steps, Step 1 and Step 2 being the same for both theorems. For the third step, we give the details in regime 2, and adapt the reasoning to regime 3-a.

5.1.

Step 1: Reduction of the set of trajectories. Recalling µ " Erωs (which is finite for α ą 1), we define

Zω

n,βn :" E " exp ´n ÿ

i"1

β n `ωi,S i ´µ1 tαě3{2u ˘¯ı (5.1)
We show that to prove Theorem 2.4 and Theorem 2.5 we can reduce the problem to the random walk trajectories belonging to Λ n,Ahn for some A ą 0 (large). For any A ą 0, we define B n pAq :" (5.9)

! pi, S i q n i"1 : max iďn |S i | ď Ah n ) ( 5
Finally we show that we can only consider a finite number of large weights. We consider Υ " Y pn,Ahnq 1 , . . . , Y pn,Ahnq ( with chosen above. Using (4.14), with probability larger 1 ´pc q ´δ {2 (with Ñ 8 as n Ñ 8) we have that

Ξ T :" ! pi, xq P 1, n ˆ ´Ah n , Ah n ; β n ω i,x ą T ) Ă Υ and thus Z pąTq n,νβn ď Z p q
n,νβn with high probability. We let L P N be a fixed (large) constant. Since |Ξ T | Ñ 8 as n Ñ 8 in probability, we have that Υ L Ă Ξ T so that, Z We now show that the contribution of Z pL, q n,νp1`η ´1qβn is negligible. Lemma 5.2. For any ε P p0, 1q and for any L P N and ρ ą 0 there exists δ L such that for all n

P ´n h 2 n log Z pL, q n,ρβn ą ε ¯ď δ L , (5.11
)

with δ L Ñ 0 as L Ñ 8.
Proof. We let ρ ą 0. Recalling the definition (3.5), and using that Pp∆ Ă Sq ď e Entp∆q , we have that Using that " oph 2 {nq and relation (5.5) of [START_REF] Berger | Entropy-controlled last-passage percolation[END_REF], we conclude the proof. ). Therefore, the proof of Theorem 2.4 and Theorem 2.5 is a consequence of relations (5.5), (5.12) and (5.13).

Proof. We detail the proof for the Regime 2. The Regime 3-a follows similarly using the results in Section 5.4 below. To keep the notation lighter we let ν " 1.

Lower bound. For any L P N we consider a set ∆ L Ă Υ L which achieves the maximum of T βn,pLq n,Ahn , resp. of r T βn,pLq n,Ahn defined below in (5.17) for Regime 3-a. We have

Z pLq n,βn ě exp ´βn Ω n,Ahn p∆ L q ¯P`S X Υ L " ∆ L ˘.
Since L is fixed, we realize that any pair of points pi, xq, pj, yq P Υ L satisfies the condition |i ´j| ě εn and |x ´y| ě εh n with probability at least 1 ´cε with c ε Ñ 0 as ε Ñ 0. In such a way, we can use the Stone local limit theorem [START_REF] Stone | On local and ratio limit theorems[END_REF] to get that PpS X Υ L " ∆ L q " n ´|∆ L | 2 `op1q e ´Entp∆Lq . In the Regime 2, in which Entp∆ L q -h 2 n {n " log n, this implies that , analogously to Proposition 3.1. Proposition 5.4. Suppose that n h 2 β n,h mpnhq Ñ ν P p0, 8q as n, h Ñ 8 and h " β 1{2 ? log n, with β ą 0. Then, for every α P p1{2, 2q and for any q ą 0, P N we have the following convergence in distribution, as n Ñ 8 n h 2 r T

Z pLq n,βn ě exp ´p1 `op1qq ! β n Ω n,Ahn p∆ L q ´Entp∆ L q ´|∆ L | 2 log n )¯, ( 5
β n,h n,qh pdq ÝÑ r T β,ν,q :" sup sPMq ! νπpsq ´Entpsq ´N psq 2β ) , (5.18)
with M q as defined in Proposition 3.1. We also have, as n Ñ 8 n h 2 r T β n,h ,p q n,qh pdq ÝÑ r T p q β,ν,q :" sup sPMq ! νπ p q psq ´Entpsq ´N psq 2β

) .

(5.19)

Moreover, we have r T p q β,ν,q pdq Ñ r T β,ν,q as Ñ 8, and r T β,ν,q pdq Ñ r T β,ν as q Ñ 8.

The proof is identical to the proof of Proposition 3.1 (cf. proof of [7, Theorem 2.7], using also that n Proof. Since r T p0q β " 0, we obtain that r T β P r0, 8q. As a by-product we also have that r T β ą 0 if and only if r T pě1q β ą 0; and in that case r T β " r T pě1q β

. Additionally, we have

W β ´1 2β ď r T pě1q β ď r T β ď ´T1 ´1 2β ¯_ 0,
with W β and T 1 defined in (2.13) and (2.6) respectively. Proposition 6.4 and Theorem 2.1 ensure that for β ą 0, W β P p0, 8q and r T 1 ă 8, showing (5.20). It remains to show that β c P p0, 8q, by observing that β Þ Ñ βW β and β Þ Ñ pβT 1 ´1{2q _ 0 are monotone functions which converge to 0 as β Ñ 0.

REGIME 3-B AND REGIME 4

In this section we prove Theorem 2.6 and Theorem 2.7. We decompose the proof in three steps (analogously to what is done in Section 5), Step 1 and Step 2 being the same for both regimes 3-b and 2. For the third step, we separate regime 3-b and regime 4, which have different behaviors. Note that in both regimes there is a constant c β ą 0 such that h n ď c ? n log n (in regime 4, we have h n ! ? n log n). Let us define here, analogously to (5.1), the recntered partition function

Zω n,βn :" E " exp ´n ÿ i"1 β n `ωi,s i ´Erω1 ωď1{βn s1 tαě1u ˘ı . (6.1)
Then, roughly speaking, we show that log Zω n,βn is of order n ´1{2 exppXh 2 n {nq, with X " r T pě1q β `1 2β in the regime 3-b (where h 2 n {n " β log n), and with X " W 1 in regime 4. In all cases, we will have log Zω n,βn " op1q (recall that in regime 3-b, r T pě1q β ă 0).

6.1.

Step 1. Reduction of the set of trajectories. We proceed as for Step 1 in Section 5: for any A ą 0 (fixed large in a moment), we define

A n :" ! pi, S i q : max iďn |S i | ď A a n log n ) . (6.2)
Then, we let Zω n,βn pA n q be the (normalized) partition function restricted to trajectories in A n . Relation (2.8) gives that, analogously to (5.4)

P

ˆˇˇl og Zω n,βn ´log Zω n,βn pA n q ˇˇě ne ´c1 A 2 log n ˙ď c 2 A ´ν1 . (

Hence, we fix A large enough so that e ´c0 A 2 log n ď n ´3. This shows that with high probability log Zω n,βn " log Zω n,βn pA n q `Opn ´2q. In such a way, in the following we can safely focus only on the partition function with trajectories restricted to A n .

6.2.

Step 2. Restriction to large weights. We now fix η P p0, 1q, small. The same Hölder inequalities as in (5.6) hold for Z ω n,βn pA n q, so that we can write, with similar notations as in (4.8)-(4.10) (the restriction to trajectories in A n does not appear in the notations)

log Zω n,βn pA n q $ ' ' & ' ' % ď 1 1 `η log Z pą1q n,p1`ηqβn `η 1 `η log Zpď1q n,p1`η ´1qβn , ě 1 1 ´η log Z pą1q n,p1´2ηqβn ´η 1 ´η log Zpď1q n,´pη ´1´1qβn . (6.4)
We used also (5.7) to be able to bound below Zpą1q n,p1´ηqβn by Z pą1q n,p1´2ηqβn (using that β n Erω1 tωď1{βnu s ! 1 when α ě 1). Then, we need to get a more precise statement than Lemma 5.1 to deal with Zpď1q n,ρβn . Lemma 6.1. For any ρ P R,

´h2 n n ¯´3α? n log Zpď1q n,ρβn P Ñ 0 , as n Ñ 8 .
Proof. We will simply control the first moment of Zpď1q n,ρβn ´1. The idea is similar to that used to obtain (4.24) and (4.31). We divide the proof into two cases: when α ă 1 so that there is no renormalization necessary in (6.1), and when α P r1, 2q.

Let us start with the case α ă 1: using that |ρ|β n ω i,S i ď |ρ| on the event tβ n ω i,S i ď 1u, we get that there exists a constant c ρ such that e

ř n i"1 ρβnω i,S i 1 tβnω i,S i ď1u ď n ź i"1 `1 `cρ β n ω i,S i 1 tβnω i,S i ď1u ˘. (6.5)
By independence, and since Ppω ą tq is regularly varying, we get that for n sufficiently large

Erβ n ω i,x 1 tβnω i,x ď1u s ď ż 1{βn 0 β n Ppω ą tqdt ď c Lp1{β n qβ α n ď cP `ω ą 1{β n ˘ď c 1 nh n ´h2 n n ¯2α . (6.6)
For the last inequality we used Potter's bound, and the definition of β n , i.e. the fact that β n " h 2 n n mpnh n q. Therefore, in view of (6.5) and using that h n ě ? n, we get that for n sufficiently large (how large depends on ρ)

E " Zpď1q n,ρβn ´1‰ ď ´1 `c1 ρ `h2 n {n ˘2α n 3{2 ¯n ´1 ď 2c 1 ρ n ´1{2 ´h2 n n ¯2α . (6.7)
This concludes the proof in the case α ă 1 by using Markov's inequality, since h 2 n {n Ñ `8. In the case α P r1, 2q, we use the expansion e x ď 1 `x `cρ x 2 for all |x| ď 2|ρ|, to get analogously to (6.5), and setting µ n :"

Erω1 tωď1{βnu s ! 1{β n , E " Zpď1q n,ρβn ı ď ´1 `ρβ n E " pω ´µn q1 tωď1{βnu ‰ `cρ β 2 n E " pω ´µn q 2 1 tωď1{βnu ‰ ¯n ď exp ´c nPpω ą 1{β n q ¯ď 1 `cn ´1{2 ´h2 n n ¯2α ,
obtaining the same upper bound as in (6.7). To obtain the above inequality, we used that

Erpω ´µn q1 tωď1{βnu s " µ n Ppω ą 1{β n q ď β ´1 n Ppω ą 1{β n q , Erpω ´µn q 2 1 tωď1{βnu s ď Erω 2 1 tωď1{βnu s ď cLp1{β n qβ α´2 n ,
where the last inequality follows similarly to (6.6). One concludes that (6.7) also holds when α ě 1, and the lemma follows by Markov's inequality.

Therefore, in view of (6.4) and Lemma 6.1, we have that for both regimes 3-b and 4

lim nÑ8 n h 2 n log ´?n log Zω n,βn pA n q ¯" lim νÑ1 lim nÑ8 n h 2 n log ´?n log Z pą1q n,νβn ¯. (6.8)
Note that in the case of regime 3-b, h 2 n {n " β log n, so the limit is that of

1 β log n log ´log Z pą1q n,νβn ¯`1 2β .
For simplicity of notations, we will consider only the case ν " 1 in the following.

6.3.

Step 3. Reduction of the main term. In both regimes 3-b and 4, we show that log Z pą1q n,βn goes to 0, and we identify at which rate: to do so, it is equivalent to identify the rate at which Z pą1q n,βn ´1 goes to 0. The behavior for regimes 3-b and 4 are different, since the main contribution to Z pą1q n,βn ´1 may come from several large weights in regime 3-b, whereas it comes from a single large weight in regime 4, as it will be reflected in the proof.

Let us define " pωq the number of pi, xq P Λ n,An " 1, n ˆ ´An , A n (with the notation A n " A ? n log n for simplicity) such that β n ω i,x ě 1, and let us denote

! pi, xq P Λ n,An ; β n ω i,x ě 1 ) " Υ :" Y pn,Anq 1 , . . . , Y pn,Anq ( , (6.9)
with Y pn,Anq i the ordered statistic, as in Section 3. We have that

Er s " ÿ pi,xqPΛ n,An Ppβ n ω i,x ě 1q ď 2An 3{2 a log n ´h2 n n ¯2α 1 nh n , (6.10)
where we used that Ppω ě 1{β n q ď ph 2 n {nq 2α pnh n q ´1 for n large enough, thanks to (2.2) and Potter's bound. Since h 2 n {n ď c log n, h n " ? n, (6.10) implies that ď plog nq 3α with probability going to 1 (we also used that 1 2 `2α ă 3α). Hence, decomposing Z pą1q n,βn according to the number of sites in Υ visited, we can write for any fixed k 0 ą 0,

k 0 ÿ k"1 U k ď Z pą1q n,βn ´1 " ÿ k"1 U k , (6.11)
with U k :" ÿ ∆ĂΥ ,|∆|"k e βnΩ n,An p∆q P `S X Υ " ∆ ˘.

In regime 3-b, the main contribution comes from one of the U k 's for some k ě 1, whereas in regime 4 only the term U 1 will contribute.

Let us now show that, with high probability, we can replace the upper bound in (6.11) by considering only a finite number of terms. For this purpose, notice that ď plog nq 3α and mint|i ´j|, pi, xq ‰ pj, yq P Υ u ě n{plog nq 10α with probability going to 1. Then, we can use the Stone local limit theorem [START_REF] Stone | On local and ratio limit theorems[END_REF] to have that for any ∆ Ă Υ P `S X Υ " ∆ ˘ď cn ´p 1 2 ´ηq|∆| e ´Entp∆q , where η ą 0 is independent of ∆ and can be chosen arbitrary small (by changing the value of the constant c).

As a consequence, using that ` k ˘ď k and ď plog nq 3α , we have for any

1 ď k 1 ď ÿ k"k 1 U k " ÿ k"k 1 ÿ ∆ĂΥ ,|∆|"k
e βnΩ n,An p∆q P `S X Υ " ∆ ˘(6.12)

ď e T βn n,An ÿ k"k 1
k n ´kp 1 2 ´ηq ď c e T βn n,An n ´k1 p 1 2 ´η1 q .

Recalling Proposition 3.1 (and the fact that h 2 n {n ď c log n) we have that T βn n,An ď C log n with probability going to 1 as C Ñ 8. Therefore, we obtain that (6.12) is Opn ´2q with probability close to 1, provided that k 1 is sufficiently large -this will turn out to be negligible, see Lemma 6.2. Hence, we have shown that with probability close to 1, we can keep a finite number of terms in (6.11). This can actually be improved in regime 4, where we can keep only one term: indeed, since in that case h 2 n {n " oplog nq, we get that for any fixed γ ą 0, T βn n,An ď γ log n with probability going to one. Hence, we get that in regime 4, we can take k 1 " 2 in (6.12) and obtain that ř k"2 U k " Opn ´3{4 q with probability close to 1, which will turn out to be negligible, see Lemma 6. with W 1 defined in (2.13).

Here also, this proves that U 1 Ñ 0 in probability, and hence so does Z pą1q n,βn ´1. 6.4. Regime 3-b: convergence of the main term. In this section, we prove Lemma 6.2.

Reduction to finitely many weights. First of all, we fix some L large and show that the main contribution comes from the L largest weights. We define

U pLq k :" ÿ ∆ĂΥL,|∆|"k e βnΩ n,An p∆q P `S X Υ " ∆ ˘, (6.15)
where Υ L " tY n,An 1 , . . . , Y n,An L u is the set of L largest weights in Λ n,An (note that Υ L Ă Υ for n large enough). Then we have that U k ě U pLq k , and

ř K k"1 U k is bounded by K ÿ k"1 ÿ ∆ĂΥL,|∆|"k ÿ ∆ 1 ĂΥ zΥL,|∆ 1 |ďK e βnΩ n,An p∆q`βnΩ n,An p∆ 1 q P `S X Υ " ∆ Y ∆ 1 ď K ÿ k"1 ÿ ∆ĂΥL,|∆|"k e βnΩ n,An p∆q P `S X Υ L " ∆ ˘ˆexp ´Kβ n M pn,Anq L " exp ´Kβ n M pn,Anq L ¯K ÿ k"1 U pLq k .
In the second inequality, we simply bounded Ω n,An p∆ 1 q by KM pn,Anq L uniformly for ∆ 1 Ă Υ zΥ L , with |∆ 1 | ď K. Then, since β n " c β plog nq{mpnh n q " c β,A plog nq{mpnA n q as n Ñ 8, we get that Kβ n M pn,Anq L is bounded above by 2c β,A KM pn,Anq L {mpnA n q ˆlog n. For any fixed ε ą 0, we can fix L large enough so that for large n we have M pn,Anq L {mpnA n q ď ε{p2Kc β,A q with probability larger than 1 ´ε. We conclude that there exists some ε L with ε L Ñ 0 as L Ñ 8 such that

0 ď K ÿ k"1 pU k ´UpLq k q ď n εL K ÿ k"1 U pLq k .
Since h 2 n {n " β log n, this proves that

lim nÑ8 n h 2 n log ´K ÿ k"1 U k ¯" lim LÑ8 lim nÑ8 n h 2 n log ´K ÿ k"1 U pLq k ¯. (6.16)
Convergence of the remaining term. We finally prove that The proof of (6.17 where for the last inequality we used Stone local limit theorem [START_REF] Stone | On local and ratio limit theorems[END_REF] (using that any two points in Υ L have abscissa differing by at least εn with probability going to 1 as ε Ñ 0) to get that P `S X Υ L " ∆ ˘" n ´|∆| 2 `op1q e ´Entp∆q uniformly for ∆ Ă Υ L . Since there are finitely many terms in the sum, we get that analogously to (5.14) In analogy with (5.17 As for the convergence of (3.8), since we have only a finite number of points, the proof is a consequence of (5.1) and (5.2) of [START_REF] Berger | Entropy-controlled last-passage percolation[END_REF] and the Skorokhod representation theorem-we use also that n h 2 n log n Ñ 1 β . Since the maximum is taken over a finite number of terms, this shows (6.18) and concludes the proof. 6.5. Regime 4: convergence of the main term. First of all, we show briefly that W β is well defined, before we turn to the proof of Lemma 6.3. One of the difficulties here is that the reduction to trajectories operated in Section 6.1 (to trajectories with max iďn |S i | ď A ? n log n) is not adapted here, since the transversal fluctuations are of order h n ! ? n log n. Therefore, we have to further reduce the set of trajectories in U 1 .

n h 2 n log ´K ÿ k"1 U pLq k ¯pdq ÝÑ max 1ďkďK r T pk,Lq β,A (6.17
Well-posedness and properties of W β . We prove the following proposition. Proposition 6.4. Assume that α P p1{2, 1q. Then for every β ą 0, W β P p0, 8q almost surely.

Proof. Recalling the definition (2.13) of W β . We fix a region D ε :" r Therefore, since 1 α ă 2, the r.h.s. of (6.19) is a.s. positive provided ε is sufficiently small. For an upper bound, we simply observe that W β ď T β ă 8 a.s.

Proof of Lemma 6.3. We denote ppi, xq :" PpS i " xq for the random walk kernel. For A ą 0 fixed and δ ą 0, we split ? n U 1 into three parts: ( . Now, let us observe that taking the limit δ Ó 0, and A Ò 8, we readily obtain that W 1 pδ, Aq Ñ W 1 (by monotonicity). Therefore, combining Lemmas 6.5-6.6-6.7, we conclude the proof of Lemma 6.3.

? n U 1 :" ÿ pi,xqPΥ e βnω i,x ? nppi, xq (6.20)
Proof of Lemma 6.5. Let us consider the event Gpn, Aq :"

! β n ω i,x ď x 2 8i for any |x| ą Ah n , 1 ď i ď n ) . (6.23)
Using this event to split the probability (and Markov's inequality), we have that, recalling the definition (6.

¯.

Using again that Ppω ě 1{β n q ď ph 2 n {nq 2α pnh n q ´1 and that ppi, xq ď e ´x2 {4i uniformly in the range considered (provided that n is large enough), we get that the first term is bounded by

1 A ´h2 n n ¯´α ? n nh n n ÿ i"1 ÿ |x|ąAhn e ´x2 {8i ď ´h2 n n ¯´α .
In the last inequality, we used that the sum over x is bounded by a constant independent of i, and also that ? n{h n Ñ 0. The first term in (6.24) therefore goes to 0 as n Ñ 8, and we are left to control PpGpn, Aq c q. A union bound gives

P `Gpn, Aq c ˘ď n ÿ i"1 `8 ÿ x"Ahn P ´βn ω i,x ě x 2 8i ¯ď n `8 ÿ k"0 2 k`1 Ahn ÿ x"2 k Ahn P ´βn ω ě 2 2k A 2 h 2 n 8n ď 2Anh n 8 ÿ k"0 2 k P ´ω ě 1 10 2 2k A 2 mpnh n q ¯, (6.25)
where we used the definition (2.2) of h n for the last inequality, with n large enough. Then, using the definition of mpnh n q and Potter's bound, we obtain that for any η ą 0 (chosen such that 1 ´2α `2η ă 0) there is a constant c ą 0 such that for n large enough

P `Gpn, Aq c ˘ď cAnh n ÿ kě1 2 k p2 2k A 2 q ´α`η 1 nh n ď c 1 A 1´2α`2η ,
where the sum over k is finite because 1 ´2α `2η ă 0. This concludes the proof of Lemma 6.5.

Proof of Lemma 6.6. Decomposing over the event

M n pδ, Aq " ! max iăδn,|x|ďAhn β n ω i,x ď 1 2 pδAq 1 4α h 2 n n ) ,
and using Markov's inequality, we get that (similarly to ( We use again that Ppω ě 1{β n q ď ph 2 n {nq 2α pnh n q ´1, and the fact that ř x ppi, xq " 1 for any i P N, to get that the first term is bounded by

e ´1 2 pδAq 1 4α h 2 n n ´h2 n n ¯2α n ? n nh n Ñ 0 as n Ñ 8 .
For the remaining term, using that β ´1 n h 2 n {n " mpnh n q, we have by a union bound that for n large enough P `Mn pδ, Aq c ˘ď δAnh n P ´ω ą 1 4 pδAq

1 4α mpnh n q ď cδAnh n ˆ`pδAq 1 4α ˘´2α 1 nh n ,
where we used Potter's bound (with pδAq 1 4α small) and the definition of mpnh n q for the last inequality (for n large). This concludes the proof of Lemma 6.6.

Proof of Lemma 6.7. The Stone local limit theorem [START_REF] Stone | On local and ratio limit theorems[END_REF] (see (2.14)) gives that, for fixed A ą 0, δ ą 0, there exists c ą 0 such that uniformly for δn ď i ď n, |x| ď Ah n ,

! w ´x2 2t ) , (6.34)
and then let ε Ó 0 -notice that we have Ă

W 1 pε, δ, Aq ď W 1 pδ, Aq ď Ă W 1 pε, δ, Aq `ε so that Ă W 1 pε, δ, Aq Ñ W 1 pδ, Aq as ε Ó 0.
We observe that a.s. there are only finitely many ω i,x in 1, n ˆ ´Ah n , Ah n that are larger than εmpnh n q " β ´1 n εh 2 n {n. This is a consequence of Markov's inequality and Borel-Cantelli Lemma. Indeed, for any K P N we have

P ´ˇˇ pi, xq P 1, n ˆ ´Ah n ,Ah n : ω i,x ě εmpnh n q ( ˇˇą 2 K ď 2 ´K p2Anh n qP ´ω ě εmpnh n q ¯ď C ε 2 ´K .
Therefore, the convergence (6.34) is a straightforward consequence of the Skorokhod representational theorem.

CASE α P p0, 1{2q

In the first part of this section we prove (2.15). In the second part, we prove the convergence (2.16). Paths cannot be at an intermediate scale. We start by showing that there exists c 0 , c, ν ą 0 such that for any sequences C n ą 1 and δ n P p0, 1q (which may go to 8, resp. 0, as n Ñ 8) and for any n ě 1

P ´Pω n,βn `max iďn ˇˇS i ˇˇP rC n ? n, δ n nq ˘ď e ´c0 C 2 n `e´c 0 n 1{2 ¯ě 1 ´cδ ν n `n´1 ´2α 4 `ε. ( 7
the sum of all weights in 1, n ˆ ´h, h . Then, we write similarly to (4.2) (we also use that Z ω n,βn ě 1, which is harmless here since no recentering term is needed) For the second term in (7.2), we partition the interval pn 3{4 , δ n nq into blocks E n,k :" r2 ´k´1 n, 2 ´knq, k " log 2 p1{δ n q, . . . , log 2 n 1{4 ´1. Exactly as above we use the large deviation estimate P `max iďn |S i | ě 2 ´k`1 n ˘ď e ´c2 ´2k n (see e.g [18, Prop. 2.1.2-(b)]), and we obtain that on the event ! @ k " log 2 p1{δ n q, . . . , log 2 n 1{4 , β n Σpn, 2 ´kn q ď c 2 2 ´2k n e ´c 2 2 ´2k n ď c 1 e ´c 2 n 1{2 . (7.7)

P ω n,
) ( 7
It now only remains to show that the complementary events of (7.4) and (7.6) have small probability. We start with (7.6). Using that β n ď 2βn{mpn 2 q for n large, we get by a union bound that P ´Σpn, 2 ´knq ą c β 2 ´2k mpn 2 q ¯.

P
Then, by Potter's bound we have that mp2 ´k`1 n 2 q ď 2 ´2k mpn 2 q since α ă 1{2 (recall mp¨q (2.1) is regularly varying with exponent 1{α). As a consequence, the last probability in (7.8) is in the so-called one-jump large deviation domain (see [START_REF] Nagaev | Large deviations of sums of independent random variables[END_REF]Thm. 1.1], we are using α ă 1 here), that is P ´Σpn, 2 ´knq ą c β 2 ´2k mpn 2 q ¯" 2 ´k`1 n 2 P `ω ą c β 2 ´2k mpn 2 q ˘.

Therefore, using again Potter's bound, we get that for arbitrary η there is some constant c such that P ´Σpn, 2 ´knq ą c β 2 ´2k mpn 2 q ¯ď cp2 2k q α`η n

´2

where we also used that Ppω ą mpn 2 qq " n ´2. Therefore, taking η small enough so that 2α ´1 `2η ă 0, we obtain that (7.8) P ´Σpn, 2 k`1 ? nq ą c β 2 2k n ´1mpn 2 q ¯. (7.9)

Then again, we notice that mp2 k`2 n 3{2 q ď 2 2k n ´1mpn 2 q (using Potter's bound, as α ă 1{2).

Hence, the last probability in (7.9) is in the one-jump large deviation domain (see [START_REF] Nagaev | Large deviations of sums of independent random variables[END_REF]Thm. 1.1]), that is P ´Σpn, 2 k`1 ? nq ą c2 2k n ´1mpn 2 q ¯ď c2 k n 3{2 P `ω ą c β 2 2k n ´1mpn 2 q Then, we also get that for any η ą 0 we have that there is a constant c ą 0 such that P `ω ą c β 2 2k n ´1mpn 2 q ˘ď cp2 2k n ´1q ´α´η , so that provided that 1 ´2α ´2η ą 0, (7.9) is bounded by a constant times

log 2 n 1{4 ÿ k"log 2 Cn
2 kp1´2α´2ηq n α´1 2 `η ď cn ´1 4 p1´2α´2ηq .

Paths cannot be at scale n conditionnaly on p T β " 0. We have shown in (7.1) that paths cannot be on an intermediate scale: it remains to prove that on the event p T β " 0, paths cannot be at scale n. For this purpose we use [5, Theorem 2.1] and [START_REF] Torri | Pinning model with heavy tailed disorder[END_REF]Theorem 1.8], which ensure that for any δ and ε ą 0 there exists ν ą 0 such that (2.16). In the following, we consider the case where β n n ´1mpn 2 q Ñ β with β ă 8. In the case β " `8, we would indeed have that p T β ą 0. The proof follows the same idea as that of [13, Thm. 1.4] (and similar steps as above), but with many adaptations (and simplifications) in our case. We focus on the case β ą 0, in which ? n βnmpn 3{2 q goes to infinity as a regularly varying function with exponent 2 α ´1 2 ´3 2α " 1´α 2α ą 0 (If β " 0, it goes to infinity faster).

P
Step 1. Reduction of the set of trajectories. Equation (2.15) (with C n " A ? log n) gives that, with P probability larger than 1´ε (conditionally on p T β " 0), we have P ω n,βn `max iďn |S i | ď A ? n log n ˘ě 1 ´e´c 0 A log n provided that n is large enough. We therefore get P ´ˇl og Z ω n,βn ´log Z ω n,βn `An ˘ˇď n ´c0 A ˇˇp T β " 0 ¯ě 1 ´ε , (7.11) where A n is defined in (6.2). Note that, provided A has been fixed large enough, we have that ? n βnmpn 3{2 q n ´c0 A Ñ 0 as n Ñ 8: we conclude that, for any ε ą 0 P ˜?n β n mpn 3{2 q ˇˇlog Z ω n,βn ´log Z ω n,βn `An ˘ˇą ε ˇˇp T β " 0 ¸ď ε , (7.12) provided that n is large enough. We will therefore focus on log Z ω n,βn `An ˘.

As in Section 6, we use the notation A n " A ? n log n " C n ? n and Λ n,An " 1, n ˆ ´An , A n .

Step 2. Truncation of the weights. We let k n :" mpn 3{2 log nq be a sequence of truncation levels, and r ω x :" ω x 1 tωxďknu be the truncated environment. Then, we have that P ´Zω n,βn pA n q ‰ Z r ω n,βn pA n q ¯" P `max We show below that lim nÑ8 W n " 0 and that V n converges in probability to 0, so that using also Lemma 7.1, we get ? n β n mpn 3{2 q log Z r ω n,βn pA n q (7.16) " ? n β n mpn 3{2 q V n `?n β n mpn 3{2 q ´pn ´1qλ n pβ n q `Wn ¯`op1q .

Before we prove the convergence of the first term (see Lemma 7.2), we show that the second term goes to 0-note that this implies that W n Ñ 0 since β n n ´1{2 mpn 3{2 q Ñ 0. We write that ˇˇpn ´1qλ n pβ n q `Wn ˇˇď pn ´1q ˇˇe λnpβnq ´1 ´λn pβ n q ˇˇ(7.17) `ˇˇn ´ÿ pi,xqPΛ n,An ppi, xq ˇˇ.

For the second term, using standard large deviation for the simple random walk (e.g. For the first term, since we have λ n pβ n q Ñ 0, we get that for n large enough ˇˇe λnpβnq ´1 ´λn pβ n q ˇˇď λ n pβ n q 2 ď ´βn mpn 3{2 q n 3{2 plog nq 2{α ¯2 , (7.19) where for the second inequality we used (7.14) (note that λ n pβ n q ď e λnpβnq ´1), together with the fact that k n ď mpn 3{2 qplog nq 2{α .

1 . 2 . 3 . 1 .

 1231 Introduction: Directed Polymers in Random Environment 1 Main results: weak-coupling limits in the case α P p0, 2q 4 Discrete energy-entropy variational problem 12 4. Transversal fluctuations: proof of Theorem 2General setting.

2. 1 .

 1 First definitions and heuristics. First of all, let us present a brief energy/entropy argument to justify what the correct transversal fluctuations of the polymer should be. Let F pxq " Ppω ď xq be the disorder distribution, and define the function mpxq by mpxq :" F ´1`1 ´1 x ˘, so we have P `ω ą mpxq ˘" 1 x .

Zpě1q n,βn ě e βnpω i 0

 0 ,x 0 ´µq P `Si 0 " x 0

  pLq n,νβn ď Z pąTq n,νβn for large n, with high probability. By using Hölder's inequality we get,

  ě exp ´p1 `op1qqT βn,pLq n,Ahn ¯. (5.14) To conclude, we use Proposition 3.1-(3.8) to obtain that T βn,pLq n,Ahn converges in distribution to T pLq 1,A , concluding the lower bound. In Regime 3-a, (5.14) is replaced by

h 2 nProposition 5 . 5 .

 255 log n Ñ 1 β in regime 3), for this reason it is omitted. To conclude, let us show that r T pěrq β defined in (2.11) is well defined. For any r ě 0 the quantities r T pěrq β are well defined and for any β ą 0 Moreover r T β ě 0, and we have r T β ą 0 if and only if r T pě1q β ą 0. Finally, there is a critical value β c " inftβ : r T β ą 0u P p0, 8q.

3 .Lemma 6 . 2 .Lemma 6 . 3 . 2 nlog ´?n U 1 ¯pdq ÝÑ W 1 ,

 36263211 It remains to show the following lemmas, proving the convergence of the main term in regimes 3-b and 4. In regime 3 (R3) (recall h 2 n {n " β log n), for any K ą 0 we have that n :" sup sPM A ,N psq"k πpsq ´Entpsq ´k 2β(, with M A defined below(5.18).Note that we have sup kě1 r T pkq β,A ă 0 in regime 3-b: this lemma proves that ř K k"1 U k goes to 0 in probability, and hence Z pą1q n,βn ´1 also goes to 0 in probability. This is needed to replace the study of log Z pą1q n,βn by that of Z pą1q n,βn ´1, and it is actually the only place where the definition of regime 3-b is used. In regime 4 (R4), we have that n h (6.14)

)

  is the restriction of r T pkq β,A to the L largest weights in r0, 1s ˆr´A, As, that is r T pk,Lq β,A :" sup sPM A ,N psq"k ! π pLq psq ´Entpsq ´k 2β In analogy with Proposition 5.4, one shows that r T pk,Lq β,A Ñ r T pkq β,A as L Ñ 8, which completes the proof.

e

  βnΩ n,An p∆q P `S X Υ L " ∆ " ÿ ∆ĂΥL,|∆|ďK exp ´βn Ω n,An p∆q ´Entp∆q ´|∆| 2 log n `opKq ¯,

  ) and Proposition 5.4, we have that for any fixed k,

7. 1 .

 1 Transversal fluctuations: proof of (2.15).

[ 18 ,PpS i ą A a n log nq ď ne ´cA 2

 182 Prop. 2.1.2-(b)]), there is a constant c ą 0 such that n ´ÿ pi,xqPΛ n,An ppi, xq " log n . (7.18)

  7, Theorem 2.7] we show that the variational problem (2.6) is indeed well defined as long as α P p1{2, 2q. In Theorem 2.4 below, we prove the second part of[START_REF] Dey | High temperature limits for p1 `1q-dimensional directed polymer with heavytailed disorder[END_REF] Conjecture 1.7]. For α P p1{2, 2q we have that T β P p0, `8q for all β ą 0 a.s. On the other hand, for α P p0, 1{2s we have T β " `8 for all β ą 0 a.s.

	Theorem 2.1 ([7, Thm. 2.4]).

  We now describe more precisely these five different regimes.

	Regime 1: transversal fluctuations of order n. Consider the case where
	β n n ´1mpn 2 q Ñ β P p0, 8s ,	(R1)
	which corresponds to having transversal fluctuations of order n. Auffinger and Louidor
	showed that, properly rescaled, log Z ω n,βn converges to p T β defined in (2.7).
	Theorem 2.3 (Regime 1, [5]). Assume α P p0, 2q, and consider β n such that (R1) holds.
	Then we have the following convergence
	1 β n mpn 2 q	log Z ω n,βn	pdq ÝÑ p T β as n Ñ 8,
	with p T

We state their results below, see Theorem 2.3 and Theorem 2.8 respectively. Our first series of results consist in identifying three new regimes for the transversal fluctuations ( ? n log n ! h n ! n, h n -? n log n, and ? n ! h n ! ? n log n), that interpolate between the Auffinger Louidor regime (h n -n) and the Dey Zygouras regime (h n -? n). β defined in (2.7). For α P r1{2, 2q, we have p T β ą 0 a.s. for all β ą 0. Regime 2: ? n log n ! h n ! n. Consider the case when β n n ´1mpn 2 q Ñ 0 and β n log n ´1mpn 3{2 a log nq Ñ 8 , (R2) which corresponds to having transversal fluctuations ? n log n ! h n ! n, see (2.2). We find that, properly rescaled, log Z ω n,βn converges to T 1 defined in (2.6)-this proves Conjecture 1.7 in [13]. Theorem 2.4 (Regime 2). Assume that α P p1{2, 2q, and consider β n such that (R2) holds. Defining h n as in (2.2), then ? n log n ! h n ! n, and we have 1 β n mpnh n q ´log Z ω n,βn ´nβ n Erωs1 tαě3{2u ¯pdq ÝÑ T 1 as n Ñ 8, (2.9)

  even at the discrete level. Our next theorem says that for β ă β c , log Z n,βn decays polynomially, with a random exponent Recalling that β n mpnh n q " h 2 n {n ! log n, we note that exp `W1 h 2 n {n ˘goes to infinity (at some random rate), but slower than any power of n.

	as n Ñ 8.
	Regime 5: transversal fluctuations of order	? n. Consider the case
								β n mpn 3{2 q Ñ β P r0, 8q ;	(R5)
	this corresponds to having transversal fluctuations h n of order	? n. Here, we state one of
	the results obtained by Dey and Zygouras, [13, Theorem 1.4].
	Theorem 2.8 (Regime 5, [13]). Assume that α P p1{2, 2q, and consider β n such that (R5)
	holds, that is β n mpn 3{2 q Ñ β P r0, 8q. Then
		? n β n mpn 3{2 q	´log Z n,βn ´nβ n E " ω1 tωďmpn 3{2 qu	‰	1 αě1	¯pdq ÝÑ 2W β pαq
	β r T β pě1q	P p´1{2, 0q.
	Theorem 2.6 (Regime 3-b, r T β " 0, β ă β c ). Assume that α P p1{2, 2q and that (R3) holds.
	Then, conditionally on t r T β pě1q	ă 0u (i.e. β ă β c ),
		1 β n mpnh n q	log ´log Z ω n,βn ´nβ n Erω1 tωď1{βnu s1 tαě1u	¯pdq ÝÑ r T β pě1q	as n Ñ 8 .
	Recalling that β n mpnh n q " h 2 n {n " β log n, we note that exppβ r T β pě1q	log nq goes to 0 as
	a (random) power β r T β pě1q	of n, with β r T β pě1q	P p´1{2, 0q.
	Regime 4:	? n ! h n !	?	n log n. Consider the case
	log nq Ñ 0 ; n ! h n ! β n mpn 3{2 q Ñ 8 and β n log n ´1mpn 3{2 a which corresponds to having transversal fluctuations ? ? n log n, see (2.2). Let (R4)
	us define
							W β :" r T β `1 2β p1q	:" sup pw,x,tqPP	! w	´x2 2βt	)	,	(2.13)
	which is a.s. positive and finite if α P p1{2, 2q, see Proposition 6.4 below.
	Theorem 2.7 (Regime 4). Assume that α P p1{2, 2q, and consider β n such that (R4) holds. Defining h n as in (2.2), then ? n ! h n ! ? n log n, and we have
			1 β n mpnh n q	log ˆ?n ´log Z ω n,βn ´nβ n Erω1 tωď1{βnu s1 tαě1u	¯˙pdq ÝÑ W 1

  Zn,βn goes to 1 with a correction of order n ´1{2 ˆeW 1 h 2 n {n , with e W 1 h 2

	• In Regimes 1 and 2, transversal fluctuations are h n "	? n log n, and Zn,βn grows faster
	than any power of n: roughly, it is of order e β p T β n in Regime 1 (for β ă 8), and of order
	e T 1 h 2 n {n in Regime 2. • In Regime 3, transversal fluctuations are h n -	?	n log n, and Zn,βn goes to infinity
	polynomially in Regime 3-a, and it goes to 1 with a polynomial correction in Regime 3-b.
	This could be summarized as Zn,βn « 1 `nβ r T β pě1q	, with β r T β pě1q	ą ´1{2: the transition
	between regime 3-a and 3-b occurs as β r T β pě1q	changes sign, at β " β c (note that β r T β pě1q
	keeps a mark of the local limit theorem, see (2.11) and (2.14)).
	• In Regime 4, n {n

)

  By convention, if t i " t i´1 for some i, then Entp∆q " `8. The set ∆ is seen as a set of points a (continuous or discrete) path has to go through: if ∆ Ă N ˆZ a standard calculation gives that Pp∆ Ă Sq ď e ´Entp∆q (∆ Ă S means that S t i " x i for all i ď |∆|), where we use that PpS i " xq ď e ´x2 {2i by a standard Chernoff bound argument. We are interested in the (discrete) variational problem, analogous to(2.6)

  then it is bounded by Erωs ă `8: this gives the first part of (4.24), using also (2.2). If α ď 1 then for any δ ą 0, for n large enough we have β n E Remaining case (α ě 3{2). We now consider the remaining case, i.e. when we do not have that n{mpnh n q In particular, we need to have that α ě 3{2, and hence Erωs ": µ ă `8. Then, we do not simply use that Z ω n,βn ě 1 to bound P ω n,βn `max iďn |S i | P B k,n ˘, but instead we use a re-centered partition function Zω

									.26)
	It remains to estimate E " . If α ą 1 " ω1 tωď1{βnu ‰ ω1 tωď1{βnu ‰ ď β p1´ηqα n for n large: by using (2.2) together with h 2 n {n ě 1, this
	gives the second part of (4.24).
	The conclusion of Lemma 4.1 follows by collecting the estimates (4.11)-(4.17)-(4.24) of
	the three terms in (4.7).
	4.2. n,βn " e ´nβnµ Z ω n,βn , so that we
	can write
	P ω n,βn `max iďn	|S i | P B k,n	˘" 1 n,βn Zω	E	"	exp	i"1 ´n ÿ	ı
				":	1 n,βn Zω	Zω n,βn `max
		P ´Z ω n,βn ě n ´1 e εn h 2 n n ¯ě 1 ´e´c{ε α´1{2´δ n	´e´cεnh 2 n {n .	(4.28)
	We postpone the proof of this lemma to the end of this subsection, and we now complete the proof of Theorem 2.2-(2.8). Lemma 4.2 gives that Zω n,βn ě n ´1 with overwhelming
	probability: using (4.2) combined with (4.27), we get, analogously to (4.5),
	P ´Pω n,β `max							n h 2 n {n ¯(4.29)
	ď P `Z ω n,βn ď n	´1˘`l og 2 pn{hnq`1 ÿ
				k"log 2 An`1

nÑ8

Ñ 0.

β n pω i,s i ´µq ¯1tmax iďn |S i |PB k,n u iďn |S i | P B k,n ˘.

(4.27)

First, we need to get a lower bound on Zω n,βn .

Lemma 4.2. For any δ ą 0, there is a constant c ą 0 such that for any positive sequence ε n ď 1 with ε n ě n ´1{2 ph 2 n {nq α´3{2`δ (this goes to 0 for δ small enough), and any n ě 1 iďn ˇˇS i ˇˇě A n h n ˘ě ne ´c1 A 2

  The last inequality holds for any fixed δ, provided that n is large enough, and comes from using Potter's bound and the relation (2.2) to get that Lp1{β n qβ α n ď c 1 Ppω ą 1{β n q ď pnh n q ´1ph 2 n {nq α`δ . Then, applying Markov and Jensen inequalities as in (4.26), we get that

	and we deal with both terms separately.
	For the first term, we use that analogously to (4.33) we have
	E	Zpď1q n,´βn ď ´1 ´βn E "	pω ´µq1 tωď1{βnu	‰	`cβ 2 n E " pω ´µq 2 1 tωď1{βnu	‰ ¯n
					ď ´1 `cLp1{β n qβ α n ¯n ď exp	´c h n	`h2 n {n	˘α`δ{2 ¯,	(4.36)
	Here, the difference with (4.33) is that we use for the second inequality that	´E" pω	μq1
	tωď1{βnu	‰	" E " pω ´µq1 tωą1{βnu	‰	ď cLp1{β n qβ α´1 n	, thanks to (1.2). Again, the second
	inequality holds for any fixed δ, provided that n is large enough. Using Markov's inequality,
	one therefore obtains that the first term in (4.35) is bounded by
	P ´Z	4.25) that pω ´µq1 tωď1{βnu 2n ¯ď exp n,6βn ď ´1 `βn E pď1q " pď1q EZ n,´βn ě e εn εnh 2 n ´c h n `h2 n {n ˘α`δ ´εn ‰ `cβ 2 n E " h 2 n 2n ¯ď exp ´´ε n pω ´µq 2 1 tωď1{βnu h 2 ¯, ‰ ¯n n 4n	(4.37)
	ď exp ´cnLp1{β n qβ α n ¯ď exp the second inequality holding provided that ε n is larger than n ´1{2 ´h2 ph 2 n {nq α`δ ¯. n n ´c h n For the second inequality, we used that E " pω ´µq1 tωď1{βnu ‰ As far as the second term in (4.35) is concerned, we find a lower bound on Z ¯α´3 2 `δ . pě1q (4.33) n,βn by ď 0 (as soon a 1{β n ě µ), and also that E " pω ´µq 2 1 tωď1{βnu ‰ ď cLp1{β n qβ α´2 restricting to a particular set of trajectories. Consider the set
					P ´log	Zpď1q n,6βn ě c 0 q 2 h 2 n n	¯ď cq	´2 n h 3 n	´h2 n n	¯α`δ	,
	which proves (4.31).
	With Lemma 4.3 in hand, and using Cauchy-Schwarz inequality as in (4.3), we get that
					P ´Z ω n,β `max
			Zpą1q n,βn{2 :" E	"	exp	´n ÿ i"1	2 β n	pω i,s i ´µq1 tβnω i,s i ą1u	¯ı
					ď p Zn,βn	˘1{2 E	"	exp	´n ÿ	´βn pω i,s i ´µq1 tβnω i,s i ą1u	¯ı1{2
								i"1
								": p Zn,βn	˘1{2 p	Zpď1q n,´βn	˘1{2 ,
	so that
								Zn,βn ě `Z	pą1q n,βn{2	˘2M Zpď1q n,´βn .	(4.34)
	Hence, we get that
								n 4n ¯,	(4.35)

n , thanks to (1.2). iďn ˇˇS i ˇˇP B k,n ˘ě 2e ´c0 2 2k h 2 n {n ¯ď p2 k q ´ν . Plugged into (4.29), this concludes the proof of Theorem 2.2-(2.8). It therefore only remains to prove Lemma 4.2. Proof of Lemma 4.2. We need to obtain a lower bound on Zn,βn , so we use Cauchy-Schwarz inequality backwards: we apply Cauchy Schwarz inequality to P ´Z ω n,βn ď n ´1 e εn h 2 n n ¯ď P ´Z pď1q n,´βn ě e εn h 2 n 2n ¯`P ´Z pą1q n,βn{2 ď n ´1{2 e εn h 2

Step 2: Restriction to large weights.

  In the second step of the proof we show that we can only consider the partition function Z We fix some δ ą 0 small, and define :" pA 2 h 2 n {nq 1´δ and also T " A 1{α h 2

	Lemma 5.1. Let ρ P R. Then,
								n h 2 n	log	Zpď1q n,ρβn	P Ñ 0, as n Ñ 8.	(5.8)
	Proof. The case ρ ą 0 is a consequence of the estimate in (4.25) and (4.26), while the case
	ρ ă 0 is a consequence of the estimate in (4.36) and (4.37)
	We can further reduce the partition function Z pą1q n,νβn to even (intermediate) larger weights
	(with ν ą 0).
								n	´p1´δq 1{2 {α
	as in (4.12): then, Hölder's inequality gives that for any η P p0, 1q	n
		log Z	pąTq n,νβn ď log Z	pą1q n,νβn ď	1	1 `η log Z	pąTq n,p1`ηqνβn `η 1 `η log Z	pp1,Tsq n,p1`η ´1qνβn .
	Then, (4.17) gives that for any fixed A ě 1, and since h 2 n {n Ñ 8, we have that for any .2) and we let Zω n,βn pB n pAqq :" E " exp ´n ÿ β n `ωi,s i ´µ1 tαě3{2u ˘¯1 BnpAq ı . (5.3) ρ ą 0, n h 2 n pp1,Tsq log Z n,ρβn
								i"1
	Relation (2.8) gives that P ´Pω
	In such a way relation (5.4) implies
				lim nÑ8	n h 2 n	log Zω n,βn " lim AÑ8	lim nÑ8	n n h 2	log Zω n,βn pB n pAqq.	(5.5)
	5.2. ω,pLq n,βn truncated to a finite number L of large
	weights, iwthL independent of n. We need some intermediate truncation steps.
	We start by removing the small weights. Using the notations introduced in (4.8 -4.10)
	and (4.32), Hölder's inequality gives that for any η P p0, 1q
		´Z	pą1q n,p1´ηqβn	¯1 1´η	´Z	pď1q n,´pη ´1´1qβn	¯´η 1´η	(5.6)
					ď Zω n,βn pB n pAqq ď ´Z	pą1q n,p1`ηqβn	¯1 1`η	´Z	pď1q n,p1`η ´1qβn	¯η 1`η ,
	We observe that the condition β n ω ą 1 implies (if µ ă 8)
		p1 ´2ηqβ n ω ď p1 ´ηqβ n pω ´µq and p1 `ηqβ n pω ´µq ď p1 `ηqβ n ω,	(5.7)
	provided n is large enough. In such a way, we can safely replace	Zpą1q n,p1´ηqβn by Z	pą1q n,p1´2ηqβn
	and	Zpą1q n,p1`ηqβn by Z

n,βn `Bn pAq ˘ě ne ´c1 A 2 h 2 n {n ¯ď c 2 A ´ν1 , uniformly on n P N. This implies that P ˆˇˇl og Zω n,βn ´log Zω n,βn pB n pAqq ˇˇě ne

´c1 1 A 2 h 2 n {n ˙ď c 2 A ´ν1 ,

(5.4)

uniformly on n P N. Let us observe that in Regime 2 and regime 3-a we have that h 2 n {n ě c β log n, therefore ne

´c1 1 A 2 h 2

n {n goes to 0 as n gets large, provided A is sufficiently large.

pą1q n,p1`ηqβn in

(5.6)

. The next lemma shows that the contribution given by log Zpď1q n,ρβn is negligible. P Ñ 0, as n Ñ 8.

Step 3: Regime 2. Convergence of the main term.

  It remains to show the convergence of the partition function restricted to the large weights.

	Collecting the above estimates, we can conclude that
	lim nÑ8	n h 2 n	log Zω n,βn pB n pAqq " lim νÑ1	lim LÑ8	lim nÑ8	n h 2 n	log Z pLq n,νβn .	(5.12)
	5.3. Proposition 5.3. For any ν ą 0, and L ą 0
		n h 2 n	log Z pLq n,νβn	pdq Ñ	# T r ν,A pLq T pLq β,ν,A	in Regime 2, in Regime 3-a,	(5.13)
	where T β,A was introduced in (3.8) and r pLq T β,ν,A is defined in (5.18) below. pLq
	One readily verifies that
	˚ν Þ Ñ T ν,A (resp. ν Þ Ñ r pLq T β,ν,A ) is a continuous function; pLq
	˚T pLq 1,A Ñ T 1,A (resp. r T β,1,A Ñ r pLq T β,1,A ) as L Ñ 8 (see Proposition 3.1, resp. Proposi-
	tion 5.4);
	˚T1,A Ñ T 1 (resp. r T β,1,A Ñ r T

β ) as A Ñ 8 (see Proposition 3.1, resp. Proposition 5.4

Step 3: Regime 3.a. Complements for the convergence of the main term.

  Using the Stone local limit theorem[START_REF] Stone | On local and ratio limit theorems[END_REF] we have that PpS XΥ L " ∆q " n ´|∆| 2 `op1q e ´Entp∆q uniformly for all ∆ Ă Υ L . Since we have only a finite number of sets, we obtain that We end here the proof of Theorem 2.5 by stating the results needed to complete Step 3 above in the case of regime 3.a. In analogy with (3.3), and in view of the local limit theorem (2.14),

		Z pLq n,βn ď 2 L exp ´p1 `op1qqT n,Ahn ¯, βn,pLq	(5.16)
	which concludes the proof of the upper bound, again thanks to the convergence proven
	in Proposition 3.1-(3.8). In Regime 3-a, using the Stone local limit theorem, we can safely
	replace T n,Ahn by r βn,pLq T n,Ahn defined below in (5.17), and also conclude thanks to Proposi-βn,pLq
	tion 5.4-(5.19).
	5.4. we define
	r T n,h :" max β n,h ∆ĂΛ n,h	β n,h Ω n,h p∆q ´Entp∆q	´|∆| 2	log n	(	,	(5.17)
	r T n,h β n,h ,p q	:" max ∆ĂΛ n,h	β n,h Ω	p q n,h p∆q ´Entp∆q	´|∆| 2	log n	(
	In the next result we state the convergence of n h 2 r T n,h and n β n,h h 2 r T n,h β n,h ,p q
								.15)
	so that T n,Ahn is replaced by r βn,pLq T n,Ahn defined in (5.17). Then the conclusion follows by βn,pLq
	Proposition 5.4-(5.19) below.
	Upper bound. We have
	Z n,βn " pLq	ÿ

∆ĂΥL

e βnΩ pLq n,qhn p∆q P `S X Υ L " ∆ ˘

  -(5.16),

	To complete the proof of (6.17) we only have to show that
	n h 2 n	log	´K ÿ k"1	U	pLq k ¯" op1q	`n h 2 n	max 1ďkďK	r T n,h β n,h ,pk,Lq	pdq Ý Ý Ñ max 1ďkďK	r T β,A . pk,Lq	(6.18)
	K ÿ k"1	U	pLq k " e oplog nq ˆexp ´max ∆ĂΥL,|∆|ďK	β n Ω n,An p∆q ´Entp∆q !	´|∆| 2	log n	)¯.
	At this stage we write
		max ∆ĂΥL,|∆|ďK	! β n Ω n,An p∆q ´Entp∆q	´|∆| 2	log n	)	" max 1ďkďK	r T n,h β n,h ,pk,Lq	,
		where	r T n,h β n,h ,pk,Lq	:"	max ∆ĂΥL,|∆|"k	β n Ω n,An p∆q ´Entp∆q !	´k 2	log n	)

  1 2 , 1sˆr´ε, εs, for ε ą 0.

	In such a way we have that
	W β ě	sup pw,t.xqPP;pt,xqPDε	w	(	´ε2 β	.	(6.19)
	We observe that
	max	w	( pdq " p2εq 1{α Expp1q ´1{α .
	pw,t,xqPP;pt,xqPDε

  The main term is the last one, and we now give three lemmas to control the three terms. There exist some c, ν ą 0 such that, for any A ą 1 and 0 ă δ ă A ´1, we get that for n sufficiently large,

	Lemma 6.6. P ˆn h 2 n	log	´ÿ pi,xqPΥ ,iăδn,|x|ďAhn	e βnω i,x ?	nppi, xq ¯ě pδAq	1 4α ˙ď cpδAq 1{2 .	(6.22)
	And finally, for last term, we have the convergence.
	Lemma 6.7. We have that
			n h 2 n	log	´ÿ pi,xqPΥ ,iěδn,|x|ďAhn	e βnω i,x ?	nppi, xq	¯pdq ÝÑ W 1 pδ, Aq ,
	with W 1 pδ, Aq :"	max pw,t,xqPP,tąδ,|x|ďA	w	´x2 2t
						"	ˆÿ pi,xqPΥ	`ÿ pi,xqPΥ	`ÿ pi,xqPΥ	˙eβnω i,x ?	nppi, xq .
							|x|ąAhn	iăδn,|x|ďAhn	iěδn,|x|ďAhn
	Lemma 6.5. There exist constants c and ν ą 0 such that for all n sufficiently large, for any
	A ą 1
				P	´ÿ pi,xqPΥ ,|x|ąAhn	e βnω i,x ?	nppi, xq ą A	´h2 n n	¯3α ¯ď cA ´ν .	(6.21)

  ´1{2 for i ě δn, so that from (6.27) we get

	On the other hand, we get that n{i ď δ n a ÿ i"δn ÿ |x|ďAhn e βnω i,x ? nppi, xq1 tβnω i,x ě1u ď	c ? δ	e βnWnpδ,Aq	n i"1 ÿ	|x|ďAhn ÿ	1 tβnω i,x ě1u . (6.30)
	Now, we have that Ppω ą 1{β n q ď ph 2 n {nq 2α pnh n q ´1 as already noticed, so that
						E " n ÿ i"1	ÿ |x|ďAhn	1 tβnω i,x ě1u	ı	ď A	ˆh2 n n	˙2α	.	(6.31)
	Overall, combining (6.28) with (6.30)-(6.31), we get that with probability going to 1 as
	n Ñ 8,
	ˇˇˇl og	´ÿ pi,xqPΥ ,iěδn,|x|ďAhn		e βnω i,x ?	nppi, xq ¯´β n W n pδ, Aq ˇˇˇď p2α `1q log	h 2 n n	.
	To conclude the proof of Lemma 6.6, it therefore remains to show that
							n
							h 2
										!	)
										|x|ďAhn, i"δn,...,n βnω i,x ąε n n h 2	ω i,x	´x2 2β n i	.	(6.33)
	In such a way, and since εh 2 n {n ě 1 for large n, we have
					n h 2 n	β n Ă W n pε, δ, Aq ď	n h 2 n	β n W n pδ, Aq ď	n n h 2	β n Ă W n pε, δ, Aq `ε.
	To prove (6.32) we need to show that
				n h 2 n	ˆβn Ă W n pε, δ, Aq	pdq ÝÝÝÑ nÑ8	Ă W 1 pε, δ, Aq :"	max pw,t,xqPP tąδ,|x|ďA,wąε
							1 c	e	´x2 {2i ď	? i ppi, xq ď c e ´x2 {2i .	(6.27)
	Since	a	n{i ě 1 for all i ď n, we get the lower bound
				n ÿ	ÿ	e βnω i,x ?	nppi, xq1 tβnω i,x ě1u ě c exp ´βn W n pδ, Aq ¯,	(6.28)
				i"δn	|x|ďAhn
	where W n pδ, Aq is a discrete analogue of W 1 pδ, Aq, that is
										!	)
						W n pδ, Aq :"	max |x|ďAhn, i"δn,...,n	ω i,x	´x2 2β n i	.	(6.29)
										βnω i,x ě1

n ˆβn W n pδ, Aq pdq ÝÝÝÑ nÑ8 W 1 pδ, Aq, (6.32) where W 1 pδ, Aq is defined in Lemma 6.6. We fix ε ą 0 and we consider Ă W n pε, δ, Aq the truncated version of W n pδ, Aq in which we replace the condition tβ n ω i,x ě 1u by tβ n ω i,x ą ε h 2 n n u, that is Ă W n pε, δ, Aq :" max

  .1)To prove it, we use a decomposition into blocks, as we did in Section 4. Here, we have to partition the interval rC n ? n, δ n nq into rC n ? n, n 3{4 q Y rn 3{4 , δ n nq (one of these intervals might be empty), obtaining

	P ω n,βn ´max iďn	ˇˇS i ˇˇP rC n	n, δnq ?	"
	P ω n,βn ´max iďn	ˇˇS i ˇˇP rC n	?	n, n 3{4 q ¯`P ω n,βn ´max

iďn ˇˇS i ˇˇP pn 3{4 , δ n nq ¯. (7.2)

For the first term, we partition the interval rC n ? n, n 3{4 q into smaller blocks D k,n :"

r2 k ? n, 2 k`1 ? nq, with k " log 2 C n , . . . , log 2 n 1{4 ´1. Let us define Σpn, hq " n ÿ i"1 ÿ xP ´h,h ω i,x

  βn ´max Cn exp ´βn Σpn, 2 k`1 ? nq ´c2 2k where for the last inequality we used a standard estimate for the deviation probability of a random walk P `max iďn |S i | ě 2 k ? n ˘ď e ´c2 2k , see for example [18, Prop. 2.1.2-(b)]. Therefore, on the event ! @ k " log 2 C n , . . . , log 2 n 1{4 , β n Σpn, 2 k`1 ?

	iďn	ˇˇS i ˇˇPrC n	?	n, n 3{4 q	k"log 2 Cn ¯ď log 2 n 1{4 ÿ	Z ω n,βn `max
			log 2 n 1{4 ÿ
		k"log 2 nq ď	c 2	2 2k	)	(7.4)
	we have that
	P ω n,β ´max

iďn |S i | P D k,n ď log 2 n 1{4 ´1 ÿ k"log 2 Cn e βnΣpn,2 k`1 ? nq P `max iďn |S i | P D k,n ď iďn ˇˇS i ˇˇP rC n ? n, n 3{4 q ¯ď log 2 n 1{4 ÿ k"log 2 Cn

e ´c 2 2 2k ď c 1 e ´c 2 C 2 n . (7.5)

  ˇˇP pn 3{4 , δ n nq ¯ď log 2 n 1{4 ÿ k"log 2 p1{δnq

	.6)
	we have
	P ω n,β ´max

iďn

ˇˇS i

  ´D k ě log 2 1{δ n , β n Σpn, 2 ´knq ą

	c 2	2 ´2k n ¯(7.8)
	ÿ
	ď
	kělog 2 1{δn

  is bounded by a constant times Similarly, for (7.4), we have by a union bound thatP ´D k P tlog 2 C n , . . . , log 2 n 1{4 u, β n Σpn, 2 k`1 ?

	ÿ	2 kp2α´1`2ηq ď cδ 1´2α`2η
	kělog 2 1{δn
		nq ą	c 2	2 2k	ď
	log 2 n 1{4 ÿ
	k"log 2 Cn

n

.

  ´Pω n,βn `max iďn |S i | P pδn, ns ˘ď e ´nν ˇˇp T β " 0 ¯ě 1 ´ε. (7.10)Therefore, we get that for any ε ą 0 and δ ą 0, combining (7.1) with (7.10), for any sequence C n ą 1, provided that n is large enough we have ?n ˘ě e ´c0 C 2 n `e´c 0 n 1{2 `e´nν ˇˇp T β " 0 ¯ď cδ ν `2ε , which concludes the proof of (2.15).

	P ´Pω n,βn `max

iďn |S i | ě C n 7.2. Convergence in distribution conditionally on p T β " 0, proof of

  (2.1) of mp¨q, so that Lpk n qk ´α n " n ´3{2 plog nq ´1. We therefore get that for n large enoughTo conclude, by Potter's bounds we get that k n ď mpn 3{2 qplog nq 2{α for n large, so that pn´1qλnpβnq ´eλnpβnq `ÿ pi,xqPΛ n,An `eβnr ω i,x ´eλnpβnq ˘ppi, xq `eλnpβnq R n

		ErR 2 n s ď	ÿ kě2	´βn k n n	¯k ď 2	´βn k n n	¯2 .
		ErR 2 n s ď	´βn mpn 3{2 q ? n	¯2	ˆplog nq n	4 α	,	(7.15)
	and the conclusion of the lemma follows by using Markov's inequality.
	Going back to (7.13), we get that
	Z r ω n,βn pA n q
	" e "
		e pn´1qλnpβnq ´1 `Vn `Wn	`eλnpβnq R n	¯,
	with
	V n :"	ÿ
						pi,xqPΛ n,An	ω i,x ą mpn 3{2 log nq ď
					2A ? log n	nÑ8 Ñ 0 ,

pi,xqPΛ n,An `eβnr ω i,x ´1˘p pi, xq and W n :" pe λnpβnq ´1q `1 ´ÿ pi,xqPΛ n,An ppi, xq ˘.

Acknowledgements We are most grateful to N. Zygouras for many enlightening discussions.

The authors acknowledge the support of PEPS grant from CNRS. Q. Berger was supported by a public grant overseen by the French National Research Agency ANR-17-CE40-0032-02. N. Torri was supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (ANR-11-LABX-0020-01 and ANR-10-LABX-0098).

where we used a union bound for the last inequality, together with the definition of mp¨q (2.1). Henceforth we can safely replace Z ω n,βn pA n q with the truncated partition function Z r ω n,βn pA n q.

Step 3. Expansion of the partition function. We write again ppi, xq " PpS i " xq for the random walk kernel, and let λ n ptq " log Ere tr ωx s. Then, expanding exp ´n ÿ

i"1

`βn ω i,S i ´λn pβ n q ˘¯" ź pi,xqPΛ n,An `1 `eβnr ω i,x ´λnpβnq ´1˘1 tS i "xu ,

we obtain

with

`eβnr ω j,x j ´λnpβnq ´1˘p n pi j ´ij´1 , x j ´xj´1 q .

Lemma 7.1. We have that for n large P ˜?n

In particular, R n Ñ 0 in probability.

Proof. Note that ErR n s " 0, so it will be enough to control the second moment of R n . Since the r ω i,x are independent and Ere βn r ω i,x ´λnpβnq ´1s " 0,

First, we have that

where S and S 1 are two independent simple random walks. Then, since β n r ω ď β n k n Ñ 0, we can write e 2βn r ω ď 1 `3β n r ω for n large, so that

To estimate the integral we used the tail behavior of Ppω ą uq (1.2) (see [8, Theorem 1.5.8]), while for the last inequality, we used that k n " mpn 3{2 log nq and the definition Hence plugging (7.18) and (7.19) into (7.17), we get that provided that A is large enough, ? n β n mpn 3{2 q ˇˇpn ´1qλ n pβ n q `Wn ˇˇď

so that the second term in (7.16) goes to 0 as n Ñ 8, proving also that W n Ñ 0 (recall also β n n ´1{2 mpn 3{2 q Ñ 0).

Step 4. Convergence of the main term. We conclude the proof by showing the convergence in distribution of the first term in (7.16) -which proves also that V n goes to 0 in probability, since β n n ´1{2 mpn 3{2 q Ñ 0.

Lemma 7.2. We have the following convergence in distribution, ? n β n mpn 3{2 q V n :" ? n

with W α 0 defined in Theorem 2.10. Proof. First of all, since β n r ω i,x ď β n k n Ñ 0 as n Ñ 8 (and using that 0 ď e x ´1 ´x ď x 2 for x small), we have that for n large

Then, we can estimate the expectation of the upper bound, using that similarly to (7.14) we have Erpr ωq 2 s ď cLpk n qk 2´α n " ck 2 n {pn 3{2 log nq. Using also that k n ď mpn 3{2 qplog nq 2{α for n large, we obtain that ? n

The proof of the lemma is therefore reduced to showing the convergence in distribution of the following term

"

where we fixed some level K ą 0 (we take the limit K Ñ 8 in the end).

First term in (7.21). First, note that the first term converges in distribution to

where ρpt, xq :" p2πtq ´1{2 e ´x2 {2t is the Gaussian kernel and Ppw, t, xq is a PPP on r0, 8q r0, 1s ˆR of intensity µpdwdtdxq " α 2 w ´α´1 1 twą0u dwdtdx. The proof of (7.22) is identical to that in [13, p. 4036], so we omit details.

Then, since W Second term in (7.21). To conclude the proof, it remains to show that the second term in (7.21) goes to 0 in probability as K Ñ 8, uniformly in n: for any K (large), we have for n sufficiently large (since K2 k ě 1). Now, we use that mp2 k`1 Kn 3{2 q ě p2 k Kq ´2{α mpn 3{2 q by Potter's bound, and also that for all k, e c 1 p2 k Kq 2 p2 k Kq ´2{α ě e 2 k K if K is large: the last probability in (7.24) is in the one-jump large deviation domain (see [START_REF] Nagaev | Large deviations of sums of independent random variables[END_REF]Thm. 1.1], we use here that α ă 1): there is a c ą 0 such that for all k ě 1 P ´n ÿ i"1 ÿ |x|ď2 k K ? n ω i,x ě e 2 k K m `2k`1 Kn 3{2 ˘ď c2 k Kn 3{2 P ´ω ě e 2 k K m `2k Kn 3{2 ˘¯ď ce ´α 2 2 k K .

The second inequality comes from Potter's bound, provided that e