Influence of disorder on DNA denaturation: the disordered generalized Poland-Scheraga model

The Poland-Scheraga model is a celebrated model for the denaturation transition of DNA, which has been widely used in the bio-physical literature to study, and investigated by mathematicians. In the original model, only opposite bases of the two strands can be paired together, but a generalized version of this model has recently been introduced, and allows for mismatches in the pairing of the two strands, and for different strand lengths. This generalized Poland-Scheraga (gPS) model has only been studied recently in the case of homogeneous interactions, then with disordered interactions perturbed by an i.i.d. field. The present paper considers a disordered version of the gPS model which is more appropriate to depict the inhomogeneous composition of the two strands (in particular interactions are perturbed in a strongly dependent manner): we study the question of the influence of disorder on the denaturation transition, and our main results provide criteria for disorder (ir)-relevance, both in terms of critical points and of order of the phase transition. Surprisingly, we find that criteria for disorder relevance depend on the law of the disorder field. We discuss this with regards to Harris' prediction for disordered systems.


Introduction
The Poland-Scheraga (PS) model has been introduced in [30] to formally study the DNA denaturation phenomenon, that is the unbinding of two strands of DNA as temperature increases. It has proven to be relevant from a quantitative point of view (see e.g. [9,10]) and has been subject to much interest from the mathematical, physical and biophysical communities (see e.g. [18,21,26,27]). In the homogeneous version of the model, i.e. when bases in each strand are all the same (for instance AAA. . . and TTT. . .), an interesting feature is that the model is solvable: it is proven to undergo a denaturation (or delocalization) phase transition, and its critical behavior can be described precisely, cf. [21,Ch. 2]. In the PS model, it is assumed that the two strands are of equal length, and that only bases from each strand with the same index can be paired. To depict DNA denaturation more accurately, the generalized Poland-Scheraga (gPS) model has been introduced more recently, where those assumptions are relaxed, see [19,20,29]. From the mathematical point of view, the gPS model can be described as a pinning model based on a two-dimensional renewal process, see [23]. Interestingly, the homogeneous version of the model remains solvable, despite having a much more complex behavior -in particular it has (in general) other critical points, corresponding to "condensation" phase transitions, see [23] and [4].
The PS and gPS models can naturally embody the inhomogeneous character of DNA. In the PS model, one introduces a sequence of random variables-referred to as disorder in statistical mechanics-describing the inhomogeneous binding energies of successive pairs. A disordered version of the gPS model has been studied recently in [5], with the introduction of a two-dimensional disorder field: the random variable of index (i, j) ∈ N 2 corresponds to the binding energy of the i-th base of the first strand with the j-th base of the second strand. In [5], the authors chose the disorder field to be i.i.d.: this assumption is relevant when using the gPS model to portray the pinning of a polymer on a inhomogeneous surface, or a directed (stretched) polymer in a random environment (in the spirit of [13,32]). However this choice is not satisfactory when describing the denaturation phenomenon between two inhomogeneous chains: the binding energy of a pair (i, j) should be a function of the i-th and j-th bases of each strand -in particular the binding energies of two pairs (i, j) and (i, k) are not independent because they have a common base.
The purpose of this paper is twofold: • study the gPS model in a setting which portrays more faithfully the pinning of two inhomogeneous polymers, as in DNA denaturation; • make progress on the understanding of disordered systems when disorder/randomness is slightly elaborate, in particular not i.i.d.   In the other one, the two strands are of different lengths and are bound on pairs (1, 1), (2,2), (6,4), (7,6), (8,7) and (12,9).
Acknowledgment. I am very grateful to Quentin Berger for the support, prolific discussions and sound advice throughout my first steps researching in the domain of polymer models.
Some notation. In the remainder of the paper, bold characters n, i , j , . . . will denote elements of N 2 (or Z 2 ), and plain characters elements of N or R. In particular we denote 1 := (1, 1), 0 := (0, 0). For r ∈ {1, 2}, the projection of any element n ∈ N 2 on its r-th coordinate will be denoted n (r) .
1.1. The generalized Poland-Scheraga model. Let τ = (τ i ) i≥0 be a bivariate renewal process: τ 0 = (0, 0), and (τ i − τ i−1 ) i≥1 are i.i.d., N 2 -valued random variables. We denote P its law, and we assume that the inter-arrival distribution satisfies for all a, b ∈ N, where α > 0, and L(·) is a slowly varying function (that is L(ux)/L(x) → 1 as x → ∞ for any u > 0, see [8]). We also assume that τ is persistent, i.e. n,m≥1 K(n + m) = 1. With a slight abuse of notation, we write τ := {τ 0 , τ 1 , τ 2 , . . .} ⊂ N 2 the set of renewal points (from now on we will omit the point τ 0 = (0, 0)). Notice that τ (r) := {τ Let ω = (ω i ) i ∈N 2 be a field of real random variables indexed in N 2 , whose law is denoted P (ω (i,j) represents the binding potential between the i-th and j-th bases of each strand). We assume that they all have the same law, and that there exists some β 0 > 0 such that for all β ∈ [0, β 0 ), λ(β) := log E[e βω 1 ] < ∞ , (1.2) (this is satisfied by bounded laws and by many unbounded laws, notably Gaussians or the product of two independent Gaussians). We also assume without loss of generality that E[ω 1 ] = 0, E[ω 2 1 ] = 1. For a fixed realization of ω (quenched disorder), we define, for β ∈ [0, β 0 ) (the disorder strength) and h ∈ R (the pinning potential), the following polymer (Gibbs) measure: for any renewal set τ ⊂ N 2 and n = (n (1) , n (2) ) ∈ N 2 , dP β,ω,q and 1, n denotes 1, n (1) × 1, n (2) ⊂ N 2 . This represents the binding of two strands with respective lengths n (1) and n (2) , and i ∈ τ if and only if the base i (1) of the first strand is paired with the base i (2) of the second strand. The polymers are constrained to be bound on the last pair n, and we give a reward (or a penalty if negative) βω i − λ(β) + h for each bound pair i ∈ τ . Notice that the term −λ(β) in the reward is present only for renormalization purposes, see for instance (1.11) below. From now on, we will drop the superscript ω in the quenched partition function and Gibbs measure to lighten notations, even though they are functions of ω.
Let us now precise our choice of disorder field. In [5], the authors studied the gPS model under an i.i.d. disorder field ω = (ω i ) i ∈N 2 . In this paper we want the disorder field to depict the inhomogeneous composition of the two strands: we pick two independent sequences ω = ( ω i 1 ) i 1 ∈N andω = (ω i 2 ) i 2 ∈N of i.i.d. random variables, whose distributions are denoted P andP respectively. These random variables are thought as being charges attached to the two strands. For each i ∈ N 2 , we fix ω i := f ( ω i (1) ,ω i (2) ) , (1.5) where f (·, ·) is a function describing the interactions between the monomers. We will write P := P ⊗P with an abuse of notation. We stress right away that ω := (ω i ) i ∈N 2 is a strongly correlated field, but that ω i and ω j are independent as soon as i , j ∈ N 2 are not aligned, i.e. are not on the same line or column: i (1) = j (1) and i (2) = j (2) .
1.2. The free energy and the denaturation transition. A physical quantity central to the study of the model is the free energy, defined in the following proposition. where n := (n, m(n)), both P(dω)-almost surely and in L 1 (P). Also, (β, h) → F γ (β, h+λ(β)) is non-negative and convex (therefore continuous on (0, ∞) × R), h → F γ (β, h) and β → F γ (β, h + λ(β)) are non-decreasing, and γ → F γ (β, h) is non-decreasing and continuous. Moreover, we have, for any 0 < γ 1 ≤ γ 2 , This proposition is analogous to [5, Thm. 1.1 and Prop 2.1] : the proof of these results is not affected by our choice of (correlated) disorder in any way whatsoever, because any trajectory of τ contributing to Z β,q n,h only involves an i.i.d. subfamily of the field ω: indeed, if i , j ∈ τ and i = j , then necessarily they are not aligned because the inter-arrivals of τ are in N 2 , hence ω i and ω j are independent. Therefore the proof of Proposition We stress that h c (β) does not depend on γ > 0, thanks to (1.7).
The critical point h c (β) marks the transition between a localized and a delocalized phase: this is the so-called denaturation (or (de)-localization) transition. Indeed, a standard calculation gives that ∂ h log Z β,q n,h = E β,q n,h i ∈ 1,n 1 {i ∈τ } : by exploiting the convexity of the free energy and Proposition 1.1, we get that whenever ∂ h F γ (β, h) exists. Therefore, for h > h c (β) we have ∂ h F γ (β, h) > 0, and in view of (1.9), there is a positive density of contacts between the two strands: they stick to each other. On the other hand, for h < h c (β) we have ∂ h F γ (β, h) = 0, and there is a zero density of contacts: the two strands wander away from one another.
1.3. The homogeneous and annealed models. The homogeneous model corresponds to the case when there is no disorder, i.e. β = 0. This model has been proven to be exactly solvable, and a fine analysis of F γ (0, h) has been performed in [23].
Moreover there are a slowly varying function L α (·) and a constant c α,γ such that (1.10) The exponent 1/ min(1, α) is often referred to as the critical exponent: it is the main quantification of the behavior of the model around its phase transition. Explicit expressions of L α are given in [23], in particular it is some constant if α > 1.
The annealed model, on the other hand, corresponds to averaging the partition function over the disorder: the annealed partition function is, for β ∈ [0, β 0 ) Here we used that for any fixed trajectory of τ the non-zero terms (βω i − λ(β) + h)1 {i ∈τ } are independent and that λ(β) = log E[e βω 1 ] < +∞ for β ∈ [0, β 0 ) (in particular this implies Z β,q n,h ∈ L 1 (P)). Therefore, the annealed free energy is Hence, we directly get from Theorem 1.2 that the annealed critical point is h a c (β) := min{h : F a γ (β, h) > 0} = 0 (and does not depend on γ). Now, a simple use of Jensen's inequality in (1.6) gives that F γ (β, h) ≤ F a γ (β, h). Moreover, we have that F γ (0, h) ≤ F γ (β, h + λ(β)) (recall that β → F γ (β, h + λ(β)) is non-decreasing). As a conclusion, we obtain the following bounds for the quenched critical point: for every (1.13) An adaptation of the proof of [21,Th. 5.2] would easily give that the second inequality is strict for every β > 0. The first inequality may or may not be strict and this is an important issue which is directly linked to disorder relevance or irrelevance.
In the rest of the paper, we will work in the case γ = 1: recall that having γ = 1 changes neither the value of the critical point h c (β), nor the homogeneous critical behavior (up to a constant factor, see inequality (1.7) and Theorem 1.2). To simplify notations, we will drop the dependence on γ in the free energy.

Presentation of the results: the question of disorder relevance
In general, going from a homogeneous model to a disordered one is a complex matter in statistical mechanics (even in the PS model, see [21,Ch. 5]). A first issue is wether the phase transition -in this paper we focus on the denaturation transition-is still present in the disordered model or not; if so, at what critical value and with what critical behavior compared to the homogeneous model. If any disorder with any strength -parametrized by β in our setting-changes the critical behavior (notably the critical exponent) of the model from the homogeneous case, disorder is said to be relevant; if a disorder of small strength does not change the critical behavior, it is said to be irrelevant.
The physicist Harris [25] predicts that disorder (ir)-relevance for a d-dimensional system can be determined from the correlation length exponent ν in the homogeneous model. If we admit that the correlation length is given by the reciprocal of the free energy, we obtain from Theorem 1.2 that ν = 1/ min (1, α). Then Harris' criterion predicts that when ν > 2/d disorder should be irrelevant, and when ν < 2/d it should be relevant (the case ν = 2/d, dubbed marginal, is much harder to treat, even with heuristic methods).
Notice that in our setting, determining the dimension d of the system is a more delicate issue than it seems: even though the disorder field ω is indexed in N 2 , it is constructed from two sequences ω andω, therefore has a 1-dimensional degree of freedom. It is not obvious if one should pick d = 1 or d = 2 in Harris' criterion. Actually we prove that there are two possible criteria for disorder (ir)-relevance depending on the law P, which correspond to Harris' prediction for each value d ∈ {1, 2}. We prove disorder irrelevance (same critical point and exponent), and disorder relevance (shift of the critical point and smoothing of the phase transition) in both cases.
We will study disorder (ir)-relevance in the case of ω andω having the same distribution P =P, and with a product interaction function f (·, ·): The condition that E[e βω 1 ] is finite for β < β 0 can be guaranteed simply by asking that ] < +∞ for β < β 0 , using that xy ≤ (x 2 + y 2 )/2. This is verified for example in the case where ω,ω are sequences of Gaussian variables, or in the case where they are bounded variables.
We will also denote m k : In particular E[ω k 1 ] = m 2 k for all k ∈ N. Recall that we assumed m 2 1 = 0 and m 2 2 = 1, in particular m 1 = 0 and m 2 = 1.

2.1.
Main results I: disorder irrelevance. Our first result is the following theorem, showing disorder irrelevance for α < 1/2, regardless of the law P.
Notice that α < 1/2 is a sufficient condition for this caim, while α ≤ 1/2 is necessary (recall that τ (1) is a univariate renewal process with inter-arrival P(τ (1) = a) = L(a)a −(1+α) , and see Proposition B.4). The upper bound in (2.2) is a direct consequence of Jensen's inequality, (1.12) and Theorem 1.2: so the interesting features are (i) and the lower bound in (2.2). This tells that, provided that β is small enough, the quenched critical point and the quenched critical exponent are the same as those given by Theorem 1.2 for the homogeneous and annealed models-which means that disorder is irrelevant.
When α > 1/2, we will state below that disorder is relevant for all disorder laws but one. Indeed, there is exactly one distribution in our setting such that disorder may also be irrelevant when α > 1/2. Let us define the distribution P ±1 as follows: (andω 1 has the same distribution). Note that this law is characterized by the identity m 4 := E[ ω 4 1 ] = 1, whereas all other laws satisfy m 4 > 1 (this follows from Cauchy-Schwarz inequality). We prove the following claim.
Let us stress that, in view of Proposition B.4, α < 1 is a sufficient condition for having τ ∩ τ terminating (while α ≤ 1 is necessary). In particular disorder with distribution P ±1 is irrelevant for all α ∈ (0, 1).

2.2.
Main results II: disorder relevance. In the two previous theorems, we stated that if m 4 > 1, (which is equivalent to P = P ±1 ), disorder is irrelevant when α < 1/2; and if m 4 = 1 (i.e. P = P ±1 ), it is irrelevant when α < 1. We now prove disorder relevance in each case for α > 1/2 and α > 1 respectively, We first focus on the shift of the critical point, starting with the case m 4 > 1.
Proposition 2.4. Assume m 4 > 1 and α > 1/2. Then there exist β 1 > 0 and some slowly varying function L 2 , such that for any β ∈ [0, β 1 ), This result is trivial when α ≥ 1: we already stated in (1.13) the upper bound h c (β) ≤ λ(β), and a Taylor expansion gives λ(β) ∼ β 2 /2 as β → 0; so the interesting feature of this proposition is when α ∈ (1/2, 1). However this upper bound is not satisfactory with regards to Theorem 2.3. We will discuss in Section 3.3 why we strongly believe that this upper bound can be improved to (almost) match the lower bound, see Remark 3.5 -actually, when ω,ω are two sequences of i.i.d. Gaussian variables, computations of Section 3.2 can be carried out exactly and give an upper bound on the shift of order L 2 (1/β)β max( 4α 2α−1 ,4) , which (almost) matches the lower bound.
When m 4 = 1 and α > 1, we prove (almost) optimal bounds on the critical point shift.
This fully covers the shift of the critical point for all disorder laws (except for the marginal cases: α = 1/2 when m 4 > 1 and α = 1 when m 4 = 1; we will discuss them at the end of this section).
With regards to the critical exponent, in the case m 4 > 1 and α > 1/2 we prove that the phase transition is smoother in the disordered model than in the homogeneous one.
This smoothing phenomenon has been first highlighted for the disordered pinning model by [24], and our proof follows the same lines.
We need an additional assumption on the disorder law P, mostly for technical reasons.
Of course we make the same assumption regardingω (we assumed P =P). Notice that Assumption 2.6 is verified when ω,ω are Gaussian sequences, and for many unbounded laws; however it does not hold for bounded disorder, in particular it does not hold for P ±1 .

Some comments on the results and the techniques of the proofs.
Criteria for disorder (ir)-relevance: m 4 > 1 vs m 4 = 1. An interesting feature of our setting is that, unlike the PS model or gPS with i.i.d. disorder, criteria on P for disorder (ir)-relevance are not the same for all disorder distributions P. We may foresee this peculiarity by looking at the correlation between rewards given by two different indices, that is E[e β(ω i +ω j ) ] − E[e βω i ]E[e βω j ], i , j ∈ N 2 (in particular those correlations appear in the proof of Theorems 2.1 and 2.2 in Section 3.2, when computing the second moment of the partition function). It is obviously 0 if i , j are on different lines and columns, and it is greater than 0 if i = j . However if i = j are on the same line or column (that is i (1) = j (1) or i (2) = j (2) ) then this correlation is 0 if and only if m 4 = 1, and it is positive otherwise, for β small (this follows from a Taylor expansion). Therefore, when m 4 > 1, the field (e βω i ) i ∈N 2 of rewards given by the disorder has strong correlations on each line and column, and the criteria for disorder (ir)-relevance in that case correspond to Harris' prediction for one-dimensional systems, i.e. the marginal regime is at the value α = 1/2. Whereas when m 4 = 1, that field is much less correlated (all its twopoint correlations are 0), and it matches Harris' prediction for two-dimensional systems, i.e. the marginal regime is at α = 1.
Comparison with the existing literature and other models. Let us compare our results to [5], where the authors studied the gPS model with an i.i.d. disorder field (ω i ) i ∈N 2 . In this setting, they prove that disorder is irrelevant as soon as τ ∩ τ terminates, and that the critical point is shifted for all α > 1 by β max( 2α α−1 ,4)+ε . Those results are the same as our Theorems 2.2 and 2.5 in the case m 4 = 1. This supports the idea that in our setting, when m 4 = 1, the field (e βω i ) i ∈N 2 is poorly correlated (even though it is not i.i.d.) and has the same influence on the system as an i.i.d. field. However when m 4 > 1, the field becomes highly correlated and the comparison with i.i.d. disorder falls appart -in particular the criteria for disorder (ir)-relevance are not the same.
Another interesting comparison is to the standard PS model (or the pinning model). The question of disorder relevance has been studied extensively in that context, see [1,6,15,21,24,31] among others. For that model, it has been proven that disorder is irrelevant if and only if the process τ ∩ τ terminates, with τ, τ two independent copies of the univariate renewal process -in particular it is irrelevant for all α < 1/2 and relevant for all α > 1/2, where we assume that the inter-arrival distribution is P(τ 1 = a) = L(a)a −(1+α) . With regards to Theorem 2.1 when m 4 > 1, and noticing that the processes τ (r) in our setting and τ in the PS model are very similar, we observe analogous criteria for disorder (ir)relevance in both systems -with a marginal value α = 1/2.
About the marginal cases. In our paper we did not fully treat the marginal cases, that is α = 1/2 when m 4 > 1 and α = 1 when m 4 = 1. By comparing our setting to the PS model, it is expected that the assumptions of Theorems 2.1 and 2.2 are optimal, and that a shift of the critical point can be proven whenever τ (r) ∩ τ (r) is persistent for m 4 > 1, respectively whenever τ ∩ τ is persistent for m 4 = 1 (notice that in [5] the authors make the same conjecture for the i.i.d. disorder). This is only conjectural for now, and we expect that a great amount of technical work is needed in both cases (for instance, bivariate renewal estimates are very complex in the case α = 1, see [3]).
About the proof of disorder relevance. Let us make some technical comments on our results for disorder relevance, starting with the shift of the critical point. Our lower bounds on the critical point shift are obtained with a coarse graining procedure, together with estimates on the fractional moments of the partition function obtained via a change of measure argument. This method has first been applied in [15] for the PS model, and was adapted to the i.i.d. gPS model in [5]. In this paper we use the same coarse-graining procedure as [5], but the change of measure argument -which in [5] relies on an i.i.d. tilt of the field ωcannot be replicated straightforwardly in the general case, because of the non-independent structure of ω (we only do it for the distribution P ±1 in Section 6, when proving the lower bound of Theorem 2.5). In the case m 4 > 1, we introduce another change of measure -a simultaneous tilt of both sequences ω andω-to prove Theorem 2.3. Interestingly enough, this change of measure relies on the correlated structure of the disorder, and does not lead to pertinent estimates in the case m 4 = 1. Moreover, it is less costly than the tilt of the field ω -for a system of size n = (n, n), we tilt 2n variables instead of n 2 . In comparison to [5] or the case m 4 = 1, this induces the appearance of a shift of the critical point when α ∈ (1/2, 1], and a greater shift when α ∈ (1, 2]. Let us stress that we have not pursued optimal upper and lower bounds on the critical point shift: the β ε in Theorems 2.3-2.5 can certainly be replaced by slowly varying functions with a more sophisticated coarse graining technique, (see [6,12] and [22,Ch. 6] for the PS model). However the proofs of Theorems 2.3 and 2.5 are already rather technical, and getting sharper bounds would have been even more laborious; so we decided to stick with these "almost" optimal lower bounds for the sake of clarity.
As far as Theorem 2.7 is concerned, a small disappointment in our result is that our proof relies heavily on Assumption 2.6, whereas one could expect to prove the same smoothing inequality for any disorder law other than P ±1 -the same way smoothing inequalities have been more recently proven for all laws in some disordered polymer models (including PS and copolymer models, see [11]). Because the disorder we consider is highly correlated, many technical difficulties appear when trying to prove a smoothing inequality, even under strong technical assumptions (e.g. bounded and symmetric disorder). We do not try to expand Theorem 2.7 under other assumptions in this paper, because we could not handle the computations without very restrictive assumptions, and even so the computations remain extremely cumbersome.
In the case m 4 = 1, we do not have any result on the smoothing of the phase transition when α > 1. Actually proving a smoothing inequality is also an issue in the gPS model with i.i.d. disorder. It is conjectured in [5,Conj. 1.5] that the gPS model model with (say Gaussian) i.i.d. disorder undergoes a smoothing phenomenon with exponent min( 2α α+1 , 4 3 ) when α > 1. Regarding our previous comments about the similarity between the i.i.d. gPS model and our setting when m 4 = 1, and comments in [5] leading the authors to their conjecture, it is reasonable to expect that a similar smoothing phenomenon occurs in our setting for disorder with distribution P ±1 , but it is purely conjectural for now.
Open questions and perspectives. Let us list here some questions on this model that are not answered yet, and perspectives for further study: (i) Improving some of our results which are not fully satisfactory, namely: proving a smoothing inequality in the case m 4 = 1 (this matter seems highly related to the gPS model with i.i.d. disorder), proving a smoothing without additional assumptions when m 4 > 1 (which seems very technical), improving the upper bound from Proposition 2.4 to match Theorem 2.3.
(ii) Dealing with the marginal cases (in particular m 4 = 1 and α = 1, because bivariate renewal processes with α = 1 are not extensively understood yet).
(iv) We also omitted the case α = 0, for two reasons: the case α = 0 sould be "strongly irrelevant", in the sens that the quenched and annealed critical points should always be equal (in the same spirit as in [2], so its should somehow be easier to prove disorder irrelevance; from a technical point of view, there are no estimates for bivariate renewals in the case α = 0 (there is no α-stable domain of attraction), and this should therefore require a separate analysis.
(iv) As stated in [23], the gPS model undergoes other phase transitions. What is the effect of disorder on those? Do they still occur in the disordered model? A conjecture on this topic can be found in [29] via heuristics and numerics, but no rigorous result has been proven yet.
(v) About the choice of disorder : we chose the interaction function f to be a product in (2.1), but we can conjecture that a generic (symmetric) function should lead to the same criteria for disorder (ir)-relevance, depending on the correlations of the field (e ω i ) i ∈N 2 . However we assumed the two sequences ω,ω (i.e. the two strands) to be independent, while it is known that two DNA strands have a strong symmetry (an A-base on one strand faces a T-base on the other, same for C-and G-bases). Therefore, a more pertinent choice of disorder field would be 9) with ω an (i.i.d.) sequence of random variables. This setting represents even more faithfully the denaturation of DNA, and it has been considered in numerical studies (see [19,20]). But it is much more difficult to handle than the one in this paper, the main reason being that the free energy is not clearly defined (even if the sequence ω is assumed i.i.d., Proposition 1.1 falls apart in that setting because the subfamily of (ω i ) i ∈N 2 involved in a trajectory τ is not always independent).

2.4.
Notation and organisation of the paper. Let us introduce some notations for the subsequent sections. C 1 , C 2 , . . . will denote some constants, and L 1 , L 2 , . . . some slowly varying functions. Unless otherwise specified, they may depend on α but will not depend on any other parameter n, h, β, . . .. Moreover L will always denote the function of the inter-arrival distribution (see (1.1)), and L α the function introduced in Theorem 1.2 for the critical behavior of the homogeneous model.
For any i , j ∈ N 2 , we will note i ↔ j if i = j or i and j are on the same line or column (i.e. if i (r) = j (r) for some r ∈ {1, 2}, recall the notation i = (i (1) , i (2) ) for all i ∈ N 2 ). We also introduce an order on N 2 : The remainder of the paper is organized into four sections. In Section 3 we show how to use a second moment method together with second moment estimates to prove both disorder irrelevance (Theorems 2.1 and 2.2) and upper bounds on the shift of the critical point (Proposition 2.4 and right-hand-side inequality in Theorem 2.5). In Section 4 we prove the smoothing inequality of Theorem 2.7 under Assumption 2.6, via a rare-stretch strategy. In Section 5, we display the coarse-graining method, and compute estimates on the fractional moments to obtain lower bounds on the shift of the critical point when m 4 > 1 (Theorem 2.3) Finally in Section 6, we adapt the change of measure argument from [5] to the distribution P ±1 to prove the same estimates on the fractional moments, thereby proving the left-hand side inequality in Theorem 2.5 (Section 6 relies on the coarse-graining procedure introduced in Section 5, the other sections are independent).
Additionally, we provide in Appendix A some computations on the partition function of the homogeneous gPS model that are needed in Section 5, and in Appendix B we collect useful estimates on bivariate renewals.

Disorder irrelevance
In this section, we choose n = (n, n) where n ∈ N (recall that we assume γ = 1). Let us define the free version of the gPS model, where the constraint 1 {n∈τ } is removed in (1.3), i.e. the endpoints are free: its partition function is defined as We claim that the constrained and free partition functions are comparable: more precisely for any α + > α and m ∈ N 2 , where C 1 is a uniform constant (see [5,Lem. 2.2]: here again the proof is not altered by our setting of disorder). In particular, the free energy is not modified with respect to (1.6): , both P(dω)-a.s. and in L 1 (P). Note that, as in [4], we defined the free partition function simply by removing the constraint {n ∈ τ }, but we mention that in the literature, the free ends may have a different entropic cost, see e.g. [5,23]. However this would not affect our results, since it has no effect at the level of the free energy.
3.1. Second moment method. Our proof of disorder irrelevance relies on the idea that, if sup n∈N E (Z β,q,free n,0 ) 2 < +∞, then Z β,q,free n,h should remain concentrated around its mean, at least in a certain regime for n, h; then the quenched and annealed free energy should remain close to each other (this idea has been exploited and turned into a proof in [1,31] for the PS model). In [28] (for the PS model again), it is roughly showed that as long as E (Z β,q,free n,0 ) 2 is of order 1, the measure P β,q,free n,0 does not differ much from P-this has also been exploited in [6]. We use this idea to obtain the following statement.
Proposition 3.1. Fix some constant C > 1, and define Then there is some (explicit) slowly varying function L 1 such that the critical point satisfies ) n∈N is bounded in L 2 (P), then n β = +∞ (provided that C had been fixed large), so h c (β) = 0; moreover there exists a slowly varying function L 2 such that for all h ∈ (0, 1), Before we prove this proposition, notice that it fully implies the non-relevance of the disorder as soon as (Z β,q,free n,0 ) n∈N is bounded in L 2 (P). We show in Section 3.2 that this holds under the assumptions of either Theorem 2.1 or 2.2 with β small, proving both theorems. Otherwise one can use (3.4) and an estimate of n β to obtain an upper bound for the shift of the critical point, which we do in Section 3.3.
(Note that it is not self-evident that E (Z β,q,free n,0 ) 2 < ∞ for any n ∈ N. We actually prove it in Section 3.2 for β < β 0 /2).
Proof. We already stated the left inequality of (3.4) with Jensen's inequality in (1.13). Let us prove the right inequality. Fix h ∈ R. We note that the sequence (E log Z β,q n,h ) n∈N is super-additive, so (1.6) and (3.2) give for any n ∈ N, Moreover we have: where we used the convexity of h → log Z β,q,free n,h , and the obvious bound Z β,q,free n,h ≥ P( τ 1 > 2n). Here we have to estimate the contact fraction of the renewal process under P β,q,free n,0 . We first do it under P. Lemma 3.2. For any ε > 0, there exist C 4 > 0 and n 0 such that for any n ≥ n 0 , 1 ] = +∞.
Let us denote A n := {|τ ∩ 1, n | ≥ C 4 f (n)}. We can estimate the contact fraction under P β,q,free n,0 with the simple inequality: so we obtain for any ε > 0 and n ≥ n 0 , where we used Paley-Zygmund inequality and Lemma 3.2. If we choose ε small enough (more precisely ε < (8C) −1 ), and n ≤ n β , this implies (where we get rid of the condition n ≥ n 0 by adjusting C 5 for finitely many terms). Going back to (3.6) and (3.7), we finally obtain the following lower bound for any n ≤ n β : where we recall f (n) := n/µ(n) if α ≥ 1 (with µ(n) either a constant, or going to +∞ as a slowly varying function in the case α = 1, E[τ 1 ] = +∞.), and f (n) := n α /L(n) if α ∈ (0, 1). If we take h = h c (β) in (3.13), then F(β, h c (β)) = 0, so we get which concludes the proof of (3.4), with ) n∈N is bounded in L 2 (P), then we can choose C such that n β = ∞, so (3.4) holds for any n ∈ N and we obviously have h c (β) = 0. Furthermore (3.13) also holds for any n ∈ N, so if we take h > 0 and n = C is a suitable constant, we finally obtain (3.5) from (3.13).
3.2. Second moment estimates. With regard to Proposition 3.1, it suffices to estimate E (Z β,q,free n,0 ) 2 to prove disorder relevance, or at least an upper bound on the shift of the critical point. To do so we introduce two independent copies τ , τ of a renewal process with law P. Proposition 3.3. For any β ∈ [0, β 0 /2) one has Z β,q,free n,0 ∈ L 2 (P) for all n ∈ N and:

15)
When m 4 = 1, one has for any β ∈ R + : Note that this proposition also applies for any sequence of indices n ∈ N 2 such that n (1) , n (2) → ∞. Plugging those estimates into Proposition 3.1, this proves Theorems 2.1 and 2.2.
Before proving Proposition 3.3, we need to introduce some new notations. Using a replica trick and Fubini-Tonelli theorem, we can write the second moment of the partition function as For any trajectories τ , τ ⊂ N 2 , τ can be written as a sequence τ = {τ 1 , τ 2 , . . .} with τ (r) k+1 for every r ∈ {1, 2}, k ∈ N, and the same holds for τ . One can rewrite (3.17) by taking the sum on i ∈ (τ ∪ τ ) ∩ 1, n (other terms are 0). We claim that there are three kind of points in τ ∪ τ contributing to this sum: • i ∈ τ ∩ τ . We will call such i double points of τ ∪ τ . (2) : that is, i is in either τ or τ , and no other point from τ ∪ τ is on the same line or column than i . We will call those isolated points of τ ∪ τ . (2) . We will call those chained points of τ ∪ τ . Let us explain the denomination chained points. Let i 1 ∈ τ ∪ τ be the first chained point for the lexical order on N 2 . Assume i 1 ∈ τ \ τ without loss of generality. Then there exists Let us now assume that there exists another Then we obviously have i 3 ∈ τ \ τ (because i 2 ∈ τ and the sequence τ has strictly increasing coordinates), and i 3 with r ∈ {1, 2} given above, since τ has strictly increasing coordinates. Moreover, this i 3 is unique (if it exists), because of the same argument that proved i 2 is unique.
• There is r ∈ {1, 2} such that for any 1 ≤ l ≤ k − 1, one has i We call σ 1 = (i 1 , . . . , i k ) a chain of points in τ ∪ τ . Note that i 1 , . . . , i k are all chained points as defined previously. Note also that this construction may lead to an infinite chain in N 2 , but is always finite if we restrict τ ∪ τ to 1, n . Furthermore, if we apply the same construction process to τ ∪ τ \ σ 1 , we can define another chain of points σ 2 satisfying the same properties ; and by repeating it again, we obtain a sequence (σ 1 , σ 2 , . . .) of chains of points in τ ∪ τ satisfying the same properties (once again this sequence is finite for τ ∪ τ restricted to 1, n , and may be infinite in N 2 ). Moreover, this sequence covers all chained points (indeed, any chained point i ∈ τ ∪ τ is only preceded by a finite number of points in τ ∪τ for the lexical order, so the construction process always reaches i ).  Figure 2. Representation of the union of two renewal sets τ , τ ⊂ N 2 . It has one double point τ 4 = τ 6 , several isolated points (τ 1 , τ 3 , τ 7 and τ 5 in lexical order), and three chains of points (τ 2 , τ 1), (τ 2, τ 4 , τ 3, τ 5 ) and (τ 8 , τ 6).
Using this construction, we can partition τ ∪ τ as stated in the following proposition.
Proposition 3.4. Let τ and τ be two subsets of N 2 such that they can be written as k+1 for every r ∈ {1, 2}, k ∈ N (same for τ ). Then the subset τ ∪ τ can be partitioned like this:

18)
where ν := τ ∩ τ , ρ is the set of isolated points of τ ∪ τ , Σ := m∈N σ m is the set of chained points of τ ∪ τ , and σ m , m ∈ N are chains of points. All the sets ν, ρ and σ m , m ∈ N are separated. Moreover, if i, j ∈ τ ∪ τ , i = j and i ↔ j, then there exists m ∈ N such that i, j ∈ σ m .
The decomposition in (3.18) comes from the construction above. The latter claim is obvious: if i ↔ j and i = j , then neither i or j can be a double point or isolated. So they are chained points, and they must be in the same chain (because of the last property of chains). This claim also ensures that disorder values (ω i ) i ∈τ ∪τ are independent from one set of the partition to another. We now have all the needed tools to prove Proposition 3.3.
Proof of Proposition 3.3. We partition (τ ∪ τ ) ∩ 1, n into ν n , ρ n , σ n,m , m ∈ N using Proposition 3.4, and we use it to separate the right hand side of (3.17) in independent products:
Notice that the particular behavior of the law P ±1 relies solely on (3.26). When σ is a chain of length two, it corresponds to computing the correlations of the field (e βω i ) i ∈N 2 , as discussed in Section 2.3.

3.3.
Upper bounds for the shift of the critical point. When we cannot bound the second moment of the partition function for all n ∈ N, we can still estimate n β with the finite volume equivalent of Proposition 3.3 (that is inequality (3.24) when m 4 > 1, and identity (3.27) when m 4 = 1): thanks to Proposition 3.1, we are able to obtain an upper bound for the shift of the critical point.
Case m 4 > 1: Proposition 2.4. We only consider the case α ∈ (1/2, 1), since the case α ≥ 1 is trivial. Let us consider (3.24): we can split the two intersections of the projections with a Cauchy-Schwarz inequality where we also used n = (n, n) and that both projections have the same law. Recall that τ (1) , τ (1) are independent univariate renewal processes with inter-arrival distribution P(τ (r) = a) = L(a)a −(1+α) , and recall that (λ(2β) − 2λ(β)) ∼ β 2 as β 0. In particular this upper bound has already been studied in [1,31] for α ∈ (1/2, 1), and gives the estimate n β ≥ L 3 (1/β)β −2 2α−1 (we do not write the details here). Therefore we obtain the upper bound from Proposition 2.4 by applying Proposition 3.1 for α ∈ (1/2, 1). Note that when α ≥ 1, estimates on τ (1) ∩ τ (1) ∩ [1, n] only give a lower bound on n β of order β −2 ; which does not lead to a better bound for the critical point than the trivial h c (β) ≤ λ(β) ∼ β 2 /2. Remark 3.5. As far as Theorem 2.3 is concerned, we strongly believe that our lower bound is of the right order (i.e. β max( 4α 2α−1 ,4) ), and that our upper bound is too rough because of the Cauchy-Schwarz inequality that we used in (3.20) to estimate the second moment (even though our estimate is sufficient to prove disorder irrelevance). Furthermore, a work in progress with Q. Berger [7] is leading us to the following claim: converges in distribution to some non-trivial random variable Z free as β 0. This convergence also holds in L 2 under some appropriate coupling.
Moreover, when ω,ω are two sequences of i.i.d. Gaussian variables, one can fully compute the contribution of chains of points to the second moment of the partition function (by iteration on the length of chains), which eventually leads to E Z β,q,free n,0 This is the same as (3.28) with an exponent of order β 4 instead of λ(2β) − 2λ(β) ∼ β 2 .
In particular it gives the expected upper bound h c (β) ≤ L 6 (1/β)β max( 4α 2α−1 ,4) on the shift of the critical point. We do not prove this claim here (because it is straightforward and purely computational, and it is useless whatsoever to treat the general case).
Case m 4 = 1. In that case we have an exact computation of the second moment (3.27), where the right hand side is the same as in the proof of [5,Prop. 3.3] for the gPS model with i.i.d. disorder. Thus we obtain the same estimate as in [5], that is n β ∼ L 7 (1/β)β − max( 2α The inequality of Theorem 2.7 is proven via a rare-stretch strategy, as done in [24] (or more recently [11]). We introduce some notations that we use in this section and the next one to lighten upcoming formulae: we write Z q n := Z β,q n,h for the (constrained) partition function with quenched disorder (the choice of parameters β and h will always be explicit), and Z n,h := Z 0,q n,h for the homogeneous (or annealed) partition function. Moreover we will denote by Z q a,b the partition function conditioned to start from a and constrained to end in b. More precisely for any 0 a ≺ b, and Z q a,b := 0 if a ⊀ b. Note that Z q a,b has same law as Z q b−a , and for any a b c d , Z q a,b and Z q c,d are independent.

4.1.
Rare-stretch strategy. The rare-stretch strategy consists in obtaining a lower bound on the partition function by considering the contribution of only one type of trajectories, which target favorable (but sparse) regions in the environment. Fix β > 0 and h ∈ R, and let A l ⊂ R l 2 , l ∈ N be a sequence of Borel sets. We will write l := (l, l), and assume that there is G ≥ 0, C ≥ 0 such that for any l ∈ N (or at least infinitely many l ∈ N): Here G stands for gain, and C for cost.  .2) (for some sequence of (A l ) l∈N ), then the following holds: Proof. We replicate here the proof of [11], but with our disorder indexed by N 2 . Fix l such that the above conditions hold, and let T i (ω), i ∈ N be the indices of blocks of size l × l on the diagonal satisfying the event A l . More precisely, let T 0 (ω) := 0, and We can give a lower bound of the partition function on the block T k l , Note that ω (T i −1)l , T i l ∈ A l for all i by definition of T i , so that Z q T i−1 l , (T i −1)l ≥ e Gl by definition of G. We also have the obvious bound Z q a,b ≥ K( b − a )e βω b −λ(β)+h for any a ≺ b. Therefore we have where the last inequality holds for any α + > α and a convenient c α + > 0, by Potter's bound (cf. [8]). We can now estimate from below the free energy, using the strong law of large numbers: where we set c α + ,β,h := log c α + −λ(β)+h. For the last inequality, we used Jensen's inequality to get that E[log(T 1 )] ≤ log(1/P(A l )) ≤ Cl. Finally, if G − (2 + α)C > 0, then it also holds for some α + > α, and the right hand side is strictly positive for l large enough, which implies F(β, h) > 0 and concludes the proof.

4.2.
Smoothing of the phase transition in β. Let us discuss the strategy of the proof first. For the PS model (with i.i.d. disorder), the method used in [24] to prove a smoothing is to fix h = h c (β), so that F(β, h c (β)) = 0 and Lemma 4.1 implies G ≤ (2 + α)C. Then one chooses a gain G close to F(β, h c (β) + u)-which matches the free energy of the model with a shifted disorder, i.e. with ω replaced by ω + u/β-and expresses the corresponding cost C with the cost of the change of measure from ω to ω + u/β (which can be estimated via a relative entropy inequality), therefore obtaining an upper bound on the free energy near the critical point. However this method doesn't apply well to our model, mostly because of the dimension of the field ω (this is also discussed in [5] for an i.i.d. disorder). A direct shift of the disorder field ω is too costly (we shift n 2 variables in a model of size n). On the other hand an i.i.d. shift of the sequences ω andω by u/ √ β -although involving only 2n variables-is not easily related to a free energy with different parameters. Therefore, we needed to adapt this method. We first prove a smoothing inequality with respect to β instead of h, using a dilation of the disorder instead of a shift, i.e. we replace the sequence ω (resp.ω) by (1 + δ) ω (resp. (1 + δ)ω). This change of measure matches the same model with disorder intensity β(1 + δ) 2 ≈ β(1 + 2δ) instead of β, and is not too costly (we change the law of 2n variables in a system of size n).
Let us introduce the "shifted" free energy F(β, h) := F(β, h + λ(β)) (i.e. if we omit the term −λ(β) in the definition of the partition function): in view of Proposition 1.1, we get that (β, h) → F(β, h) is convex, so the critical line β → h c (β) := inf{h : F(β, h) > 0} = h c (β) − λ(β) is concave. Actually it is decreasing and continuous (recall that the upper bound in (1.13) is strict for β > 0), so one can consider the inverse map of β → h c (β), that we denote h → β c (h): for each h > 0, the value β c (h) is the critical value for the map β → F(β, h) corresponding to the localization transition. One can therefore consider the transition as β varies, and the next proposition tells this phase transition is smooth, i.e. for fixed h, the growth of β → F(β, h) close to β c (h) is at most quadratic.
Proposition 4.2. Under Assumption 2.6, for any h > 0, there exists c h > 0 such that for any δ ∈ (0, 1), one has (4.8) Proof. For any δ > 0, we define P l,δ the law of the disorder in 1, l dilated by (1 + δ) on each coordinate (i.e. ω 1,l is replaced with (1 + δ) ω 1,l , same forω 1,l ). We denote P δ the infinite product law. Note that H( P l,δ |P) = lH( P 1,δ |P), and that there is some c > 0 such that H( P 1,δ |P) ≤ c δ 2 by Assumption 2.6. For any β > 0, h ∈ R and l ∈ N, let us define where Z q l := Z β,q l ,h+λ(β) , so that lim l→+∞ 1 l log Z q l := F(β, h). The set A l is chosen so that the gain (as defined for Lemma 4. 2 , h is exactly the free energy of the gPS model with partition function Z q l , where we changed the disorder law from P to P δ (because multiplying ω andω by (1 + δ) is the same thing as multiplying β by (1 + δ) 2 ). Thus 1 l log Z q l converges P δ -a.s. to F β(1 + δ) 2 , h when l → ∞, so we obviously have that (4.10) Now we can estimate P(A l ) via a standard relative entropy inequality, which gives (4.11) We therefore get that P(A l ) ≥ e −2/e 2 e −2cδ 2 l , for l large enough (so that P l,δ (A l ) ≥ 1/2). Therefore, for l large (how large depends on δ), we get that This concludes the proof by simply recalling that F is non-decreasing in its first coordinate, so

Conclusion: smoothing of the phase transition in h.
Once we have the smoothing inequality with respect to β of Proposition 4.2, we are able transcribe it to a smoothing in h using the convexity properties of F(β, h). We now conclude the proof of Theorem 2.7, thanks to Proposition 4.2.
Proof of Theorem 2.7. We fix β > 0, and we take t > 0 small enough such that h c (β)+t < 0.
Recall that h c (·) is a concave, non-increasing, continuous function: there exists β t ∈ (0, β) such that h c (β t ) = h c (β) + t. Hence, where u = β − β t > 0. Using Proposition 4.2 and the fact that β c ( h c (β t )) = β t , we have Note that β t β > 0 as t 0, so the factor c hc(βt) β −2 t is bounded by a constant that depends only on β, uniformly in t sufficiently small. Using that h c (·) is concave, we also have where h c (· + ) is the right derivative of h c , and the second inequality holds for any t sufficiently small (because h c (β + t ) < 0 as soon as β t > 0, and it decreases as t 0). We deduce that t ≥ c β u for some c β > 0, and plugging it in (4.15), we finally obtain where c β > 0 depends only on β. This concludes the proof of Theorem 2.7.

5.
Disorder relevance: shift of the critical point when m 4 > 1 In this section, we prove Theorem 2.3, that is a the lower bound for the shift of the critical point when m 4 > 1. We will discuss the case m 4 = 1 afterwards in Section 6.

5.1.
Coarse-graining and fractional moment method. Our proof is based on a fractional moment method, introduced in [15] for the original PS model, and slightly adapted to the gPS model with independent disorder in [5]. The first part of the proof (the coarsegraining procedure) is identical to that of [5], but the different estimates are specific to our setting.
Recall the notation Z q n := Z β,q n,h from Section 4. Let us define for any n ∈ N 2 (note that we do not assume n = (n, n) in this section) the fractional moment of the partition function: where η ∈ (0, 1) is a constant we will fix later on (notice that A n < ∞ because Z β,q n,h ∈ L 1 (P)). For any k ∈ N we set k := (k, k), and for n k , we decompose the partition function Z q n according to the first point n − i of τ which lies in the square n − k + 1, n (in particular 0 i ≺ k ), and the point n − i − j before: in particular it is the last point of τ which isn't in the previous square, so it can only be in one of the three boxes 0, n (1) − k × 0, n (2) − k , n (1) − k, n (1) × 0, n (2) − k or 0, n (1) − k × n (2) − k, n (2) . This decomposition gives us for any n k , where we write for any s ∈ {1, 2, 3}, where Z ω n−i ,n is defined in (4.1), and the sets D s k ,n , s ∈ {1, 2, 3} are defined as follow (with We also define D s k ,∞ = ∪ n∈N 2 D s k ,n (which means we drop the condition i + j n). Recall that Z q n−i ,n and Z q i have same law, and that Z q n−i −j , e βω n−i −λ(β)+h and Z q n−i ,n are independent. Using this together with the definition of A n and the standard inequality ( i a i ) η ≤ i a η i for any η ∈ (0, 1) and a i ≥ 0, i ∈ N, we obtain where for any s ∈ {1, 2, 3}, and c β,h,η := E[e (βω 1 −λ(β)+h)η ] = e λ(ηβ)−ηλ(β)+ηh . Note that c β,h,η is uniformly bounded for β, h small and 0 ≤ η ≤ 1, so we can bound it by a constant C 1 . As in [5], the proof of Theorem 2.3 relies on the following claim.
Proof. The proof is straightforward. Define A := sup{A i , i (1) < k or i (2) < k}. By Jensen's inequality, one obviously has A i ≤ E[Z q i ] η ≤ e ηh min(i (1) ,i (2) ) , because there are at most min(i (1) , i (2) ) renewal points in 1, i . Thus we have A ≤ e ηhk . Using the decomposition (5.5), (5.6) of A n and ρ 1 + ρ 2 + ρ 3 ≤ 1, we deduce (by induction) A n ≤ A for any n ∈ N 2 . By applying Jensen's inequality, we conclude Proof of Theorem 2.3. We now assume m 4 > 1 and α > 1/2. We fix h as in Theorem 2.3: where ε > 0 is arbitrarily small, but fixed. Our goal is to choose η ∈ (0, 1) and k ∈ N such that ρ 1 , ρ 2 and ρ 3 (which is symmetric to ρ 2 ) are small, so that Lemma 5.1 implies F(β, h) = 0 and h c (β) ≥ h. First we pick which is the correlation length of the annealed system (actually we take the integer part of it, but we omit to write it for clarity purpose). Notice that k → ∞ as h → 0, and recall that Theorem 1.2 allows us to write k = L α (1/h)h −1/α with L α a slowly varying function when α ≤ 1; and k ∼ c α h −1 for some c α > 0 as h 0 when α > 1 (here we slightly changed the notations from the theorem, to lighten upcoming computations).
Note that, if η is picked such that (2 + α)η > 2, then we have Therefore, (2) ). Similarly, Thus, Let us denote this sum S. Because ρ 3 is symmetric to ρ 2 , and because ρ 1 and ρ 2 are bounded by the same sum S (up to the slowly varying function, which doesn't change the behavior of the sum), Theorem 2.3 will be proven as soon as we show that S can be made small, by applying Lemma 5.1. For that we need some precise estimates on A i for i ≺ k . We will handle A i differently depending on whether i is small or not, using Lemmas 5.2-5.3 below.
Lemma 5.2. There exist constants C 2 > 0, h 1 > 0 and a slowly varying function L 5 such that for any h ∈ (0, h 1 ) and n with 1 ≤ n ≤ 1/F(0, h), one has where Z n,h is the partition function of the homogeneous model with parameter h.
Note that A i ≤ (EZ q i ) η = (Z i ,h ) η because of Jensen's inequality, therefore this lemma gives us a first bound on A i , valid for all i k if α = 1. When α = 1 we only state it for i ≤ k 1−ε 2 to avoid some technicalities in the proof.
We will use Lemma 5.2 to handle the small values of i . When i is not too small, and under the assumptions m 4 > 1, α > 1/2, this bound can be improved. Lemma 5.3. Assume that m 4 > 1 and α > 1/2. If ε has been fixed small enough, then there exist some constants C 3 , C 4 , C 5 > 0 such that for any k (1−ε 2 ) ≤ n ≤ 2k, one has We delay the proof of Lemma 5.2 in Appendix A (because it only concerns the homogeneous gPS model), and we prove Lemma 5.3 later in this section. We now have all the tools we need to finish the proof. Fix ε > 0 sufficiently small for Lemma 5.3 to apply, then define η depending on α > 1/2.
Case α ∈ (1/2, 1). Choose η such that In particular (2 + α)η > 2 so (5.12) and (5.14) hold and we only have to control the sum S defined in (5.14). We introduce the notation u k := k (1−ε 2 ) , and we part the sum in two: To bound S 1 , we use Lemma 5.3 to estimate A i .
For S 2 , we bound the first factor uniformly in i (2) (note that (k − i (2) ) ≥ k/2 for i < u k ), and we estimate A i with Lemma 5.2.
where we used (2 − α)η ∈ (1, 2), and we recall u k = k (1−ε 2 ) (note that we had to write separately the term i = 0). The exponent in the denominator of the first term can be written (4 − 2ε 2 + ε 2 α))η − 4 + 2ε 2 which is positive because of (5.17), and the other exponent is also positive. Thus S 2 is also small for k large (i.e. β small).
For S 2 , we bound the first factor uniformly in i (2) and estimate A i with Lemma 5.2.
The exponent in the denominator of the first term can be written 2 + α + 1−ε 2 min(α,2) η − (4 − 2ε 2 ), which is positive. Finally S 2 vanishes too as k → ∞, and this concludes the proof of Theorem 2.3 in the case α > 1 with Lemma 5.3.
We conclude the proof of Lemma 5.3 by applying this result with u := c|δ|β 2 = c n −1/2 β 2 (which decays to 0 as β 0). Case α ∈ (1/2, 1]. We apply Lemma 5.4 to (5.38), i.e. with u = c n −1/2 β 2 . Notice that, thanks to (5.36), we have Provided that ε is sufficiently small (so that ε 2α−1 2α > 2c 0 ε 3/2 ), we can choose α − sufficiently close to α (depending on ε) so that the exponent is negative; thus u n α − goes to infinity as a power of β. In particular the second term in (5.42) decays much faster than the first one (which decays at most polynomially in β, recall (5.36)). Hence, Lemma 5.4 and (5.36) give for any β sufficiently small, In this section we prove the lower bound in Theorem 2.5, and in particular we discuss how the estimate of the fractional moment (i.e. Lemma 5.3 when m 4 > 1) have to be adapted. Note that we can reproduce exactly the first part of the above proof, namely the coarse-graining procedure whose core is Lemma 5.1, and we can use Lemma 5.2 as it is.
On the other hand, the change of measure argument needs important adaptation. In the case of an i.i.d. disorder, what plays the role of Lemma 5.3 is [5,Prop. 4.2]: there, the estimates of fractional moments A i use an i.i.d. tilt of the disorder, but only along an extended diagonal, i.e.
The width n is chosen depending on α > 1, so that the renewal process τ is very unlikely to deviate from the diagonal by more than n (see [5,Thm. A.5], or [6,Thm. 4.2] for a more general statement): (notice that |J n | ≤ 2n (1) n n (1) n (2) when n (1) ≈ n (2) ). In our setting the nonindependent disorder adds some technicalities to this method, but they can be handled if we restrain ourselves to P ±1 . We prove the following result, which plays the role of Lemma 5.3 in the case m 4 = 1.
Proposition 6.1. Assume α > 1, m 4 = 1, recall k = 1/F(0, h), and define n as in (6.2). Then there exist h 0 > 0 and L 1 such that for any h ∈ (0, h 0 ) and √ k ≤ n (1) ≤ k, n (1) ≤ n (2) ≤ n (1) + n , one has Note that these are exactly the same estimates on A n as in [5,Prop. 4.2]. Once plugged in the computations of ρ 1 , ρ 2 and ρ 3 from Lemma 5.1, they give the same lower bound for the shift of the critical point as in [5,Thm 1.4], which is This proves the left inequality in Theorem 2.5. We do not write the details here, because once we have the estimates on A n from Proposition 6.1, the computations of ρ 1 , ρ 2 and ρ 3 are the same as in [5] and do not depend on the setting of disorder.
Proof of Proposition 6.1. This follows the same scheme as [5,Prop. 4.2]. We define the same change of measure on J n as [5, (4.18)], that is dP n,δ dP (ω) : Applying Hölder's inequality to A n similarly to (5.30), we have Let us fix δ := −(n (1) n ) − 1 2 (1+ε 3 ) < 0 -this is the same as in [5] where we added the power (1 + ε 3 ) to avoid technicalities. We claim the following: Lemma 6.2. There exists C 2 > 0 and δ 1 > 0 such that if δ 2 n (1) n ≤ δ 1 , then Proof of Lemma 6.2. The lower bound follows directly from Jensen's inequality, so let us focus on the upper bound. For all 1 ≤ i ≤ n (1) , we define J n (i) := {j ∈ 1, n (2) ; (i, j) ∈ J n } , and σ i = σ i (ω) := j∈Jn (i)ω j , (6.8) in particular |σ i | ≤ |J n (i)| ≤ 2 n . Then, computing first the expectation conditionally on ω, we have where we used that ω is a sequence of independent variables, and is independent fromω (and that E[e x ω i ] = cosh(x) for all x). Notice that for all x ∈ R, one has cosh(x) ≤ e x 2 2 : we therefore have by Cauchy-Schwarz inequality. Here σ i is a sum of |J n (i)| i.i.d. bounded variables, so converges in distribution to some standart gaussian Z ∼ N (0, 1) as |J n (i)| → ∞.
Proof of Lemma 6.3. The left inequality is a straight consequence to Jensen's inequality, so we focus on the upper bound. We introduce some notations specific to this lemma. Let r := |τ ∩ 1, n | and denote (a l , b l ) = τ l , 1 ≤ l ≤ r (in particular (a r , b r ) = τ r = n). Moreover for all 1 ≤ l ≤ r, denote J n (a l ) = c l , d l . Then we can write where we parted the sum according to indices ofω before and after b l . Let us define X 0 = Y 0 = 0 and for all 1 ≤ t ≤ r, so that n (1) i=1 σ iσ i = X r + Y r , and by Cauchy-Schwarz inequality, E exp 2δ tanh(β) (6.26) We only treat the first factor, the second one being symmetric. We define F 0 the trivial σ-algebra, and for all 1 ≤ t ≤ r, with b r+1 := n (2) + 1: it is then easily checked that (X t ) 0≤t≤r is a (F t ) 0≤t≤r -martingale. We also define a "truncated" version of X: for all w > 0, X (w) 0 := 0 and for all 1 ≤ t ≤ r, we set . (6.28) Notice that (X (w) t ) 0≤t≤r is also a (F t ) 0≤t≤r -martingale, and it has bounded increments: |X t−1 | ≤ w n log(n (1) ) for all 1 ≤ t ≤ r. Therefore we deduce from the Azuma-Hoeffding inequality that for all u, w > 0, P |X (w) r | > u r n log(n (1) ) ≤ 2 exp − u 2 2w 2 . (6.29) Moreover we have so by Hoeffding's inequality, and because |J n (i)| ≤ 2 n and r ≤ n (1) , We deduce from (6.31) and (6.29) that for all u, w > 0, P |X r | > u r n log(n (1) ) ≤ 2 exp − u 2 2w 2 + n (1) exp − w 2 4 log(n (1) ) , (6.32) in particular with , P |X r | > u r n log(n (1) ) ≤ (2 + n (1) ) exp − u log(n (1) ) 2 √ 2 ≤ exp(−Cu), (6.33) where C > 0 is a constant sufficiently small such that this inequality holds for all u ≥ 1 and n (1) ∈ N. Hence, (6.33) implies that |Xr| √ r n log(n (1) ) is dominated under some coupling by C + Z, with C a constant and Z an exponential random variable with parameter C. In particular, E exp 4δ tanh(β)X r ≤ E exp 4|δ|β C + r n log(n (1) )Z , (6.34) and this can be made aritrarily small if |δ|β r n log(n (1) ) ≤ |δ| n (1) n log(n (1) ) is small, which concludes the proof of the lemma. For the first sum, (recall α − < α ≤ 1) we use Proposition B.1 to bound P τ j = n by a constant times jK( n ), both for α < 1 and α = 1 (for the latter, notice that b j n ). We obtain n α − j=1 e −ju P τ j = n ≤ C 1 K( n ) u 2 where we used a Riemann-sum approximation of the last sum to get that it is bounded by a constant times R + xe −x dx = 1 for any u ≤ u 0 . For the terms j > n α − , we simply bound j from below: +∞ j= n α − +1 e −ju P τ j = n ≤ e −u n α − P(n ∈ τ ), (A.3) and the proof is complete.
Those rough estimates do not apply when α ∈ (0, 1), since we do not have uniform bounds on h|τ ∩ 1, n |. We follow the line of proof of [15,Lem. 4.1] for the PS model. A first step in the proof consists in dealing with the indicator function in the partition function, to compare it with its free counterpart, see (A.9) below. Let us define T n := i ∈ 1, n ; i ≤ 1 2 n , (A.5) the lower left half of the rectangle 1, n , where we recall that · is the L 1 norm on N 2 . Because, conditionally on n ∈ τ , the time-reversed process τ in 1, n \ T n starting from n has same law as τ in T n , the partition function is bounded with a Cauchy-Schwarz inequality by Z n,h = E e h |τ ∩ 1,n | n ∈ τ P n ∈ τ ≤ E e 2h |τ ∩Tn | n ∈ τ P n ∈ τ , (A.6) (if n is even the anti-diagonal { i = n /2} is counted twice, but the upper bound still holds). Let us define X n := sup{i ∈ T n ∩ τ } the last renewal point in T n (we take the supremum for the natural order on τ ⊂ N 2 : recall that τ is strictly increasing on both coordinates). Because |τ ∩ T n | and 1 {n∈τ } are independent conditionally to {X n = i }, we can write: E e 2h|τ ∩Tn | n ∈ τ = i ∈Tn E e 2h|τ ∩Tn | X n = i P X n = i n ∈ τ . (A.7) then we can use the following Lemma, which is proven afterwards.
Lemma A.1. Assume α ∈ (0, 1). There exists C 4 > 0 such that for any n ∈ N and i ∈ T n , P X n = i , n ∈ τ ≤ C 4 L( n ) −1 n −(2−α) P X n = i . so Lemma 5.2 will be proven once we show that the above expectation is bounded by a constant, uniformly for n ≤ 1/F(0, h). Notice that τ = ( τ i ) i≥1 is a renewal process on N, so we may write the upper bound E e 2h |τ ∩Tn | ≤ E exp 2h which is the partition function of a homogeneous PS model with underlying univariate renewal τ , which easily verifies P( τ 1 = k) ∼ L(k)k −(1+α) as k → +∞. The right side of (A.10) has already been studied in [15,Lemma 4.1] when α ∈ (0, 1), and we therefore get that the expectation is bounded by a constant uniformly in n ≤ 1/F(0, h).
Proof of Lemma A.1. Fix n, i ∈ N 2 . Recall and the definitions of T n in (A.5) and X n , and that we assumed α ∈ (0, 1). We write: P X n = i , n ∈ τ = P i ∈ τ P τ 1 > 1 2 n − i , n − i ∈ τ = P i ∈ τ j ∈Q n i P τ 1 = j P n − i − j ∈ τ , (A.11) with Q n i := j ∈ N 2 ; j n − i , j > 1 2 n − i . (A.12) Then, we can use Proposition B.2 to get that P n − i − j ∈ τ is bounded by a constant times L( n − i − j ) −1 n − i − j −(2−α) . We get that the sum in (A.11) is bounded by a constant times j ∈Q n i , j ≤ n /4 P τ 1 = j where we decomposed the sum according to whether j ≤ n /4 or not. We used that if j ≤ n /4 then n − i − j ≥ n /4, and if j > n /4 then we can bound P(j ∈ τ ) uniformly thanks to (1.1), and we wrote := n − i − j .

Appendix B. Some properties of bivariate renewals
We provide here some estimates on the bivariate renewal τ that we use (recall its interarrival distribution (1.1)). They can be found in [5,Appendix A] or in [3] in a more general setting, and rely on the fact that τ is in the domain of attraction of a min(α, 2)-stable distribution. We define the scaling sequence a n , a n := ψ(n)n 1/ min(α,2) , (B.1) where ψ is some slowly varying function (we do not detail it here, see [3,5] ; notice that if α > 2, ψ is a constant). We also define the recentering sequence b n , b n := 0 if α ∈ (0, 1), b n := n µ(a n ) if α = 1, b n := n µ if α > 1.
We stress that this upper bound is sharp when n is close to the diagonal, we refer to [3,] for the precise statements. Also, notice that n → µ(n)a n/µ(n) is regularly varying with index 1/ min(α, 2).
Let us also recall some results on intersections of renewals, either univariate or bivariate. We omit the marginal cases (α = 1/2 for univariate renewals, and α = 1 for bivariate) for the sake of simplicity, and because we do not treat these cases in the present paper.