The random pinning model with correlated disorder given by a renewal set

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent α > 0, when the correlated sequence is given by another independent renewal set with loop exponent α̂ > 0. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case α̂ > 2 (summable correlations), disorder is irrelevant if α < 1/2 and relevant if α > 1/2, which extends the Harris criterion for independent disorder. The case α̂ ∈ (1, 2) (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for α > 1/α̂, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case α̂ ∈ (0, 1) is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.

in the caseα > 2 (summable correlations), disorder is irrelevant if α < 1/2 and relevant if α > 1/2, which extends the Harris criterion for independent disorder. The caseα ∈ (1, 2) (nonsummable correlations) remains largely open, but we are able to prove that disorder is relevant for α > 1/α, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the caseα ∈ (0, 1) is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.

Introduction
The goal of this paper is to study the phase transition of the pinning model in presence of a correlated disorder sequence built out of a renewal sequence. We first present the general set-up of pinning models before introducing our specific model. For a review on pinning models, we refer to the three monographs [Gia07,Gia11,dH09] and references therein. In this paper we write N = {1, 2, . . .} and N 0 = {0, 1, 2, . . .}.

General set-up
The pinning model provides a general mathematical framework for studying various physical phenomena such as the wetting transition of interfaces, DNA denaturation, or (de)localization of a polymer along a defect line. This statistical-mechanical model is formulated in terms of a Markov chain (S n ) n∈N 0 which is given a reward/penalty ω n (depending on the sign) when it returns to its initial state 0 at time n.
Let us denote by τ = (τ n ) n∈N 0 the sequence of return times to 0, whose law is denoted by P. It is a renewal sequence starting at τ 0 = 0, and we assume that the inter-arrival law satisfies (1.1) K(n) := P(τ 1 = n) = L(n) n −(1+α) , α > 0, n ∈ N, where L is a slowly varying function whose support is aperiodic, that is, gcd{n 1 : L(n) > 0} = 1. We also assume that the renewal process is recurrent, that is P(τ 1 < ∞) = n 1 K(n) = 1 (otherwise it is said to be transient). By a slight abuse of notation, we shall use τ to refer to the set {τ k } k∈N 0 and write δ n = 1 {n∈τ } .
Independently of τ , we introduce a disorder sequence, that is a sequence of real valued random variables ω = (ω n ) n∈N 0 whose law is denoted by P.
The object of interest is the sequence of Gibbs measures, also called polymer measures, defined by: (1.2) dP n,β,h dP = 1 Z n,β,h exp n k=1 (h + βω k )δ k δ n , n ∈ N, β 0, h ∈ R, where (1.3) Z n,β,h = E e n k=1 (h+βω k )δ k δ n is the quenched partition function, h is called a pinning strength or chemical potential, and β is the inverse temperature.
The free energy of the model is defined by (1.5) h c (β) = inf{h : F (β, h) > 0}, ν q (β) = lim h hc (β) log F (β, h) log(h − h c (β)) , when the limit exists. The critical curve separates the two phases whereas the critical exponent indicates how smooth the transition is between them.
1.1.1. Disorder relevance One reason for the success of this model is the solvable nature of the homogeneous case, which corresponds to the choice β = 0 and which is treated in detail in [Gia07]. For the moment, we recall that h c (0) = 0 and ν hom := ν q (0) = max(1, 1/α), see [Gia07,Theorem 2.1].
An important challenge in statistical mechanics is to understand the effect of quenched impurities or inhomogeneities in the interaction on the mechanism of the phase transition. This can be done by comparing the critical features of the quenched model to that of the annealed model, which is defined by (1.6) dP a n,β,h dP = 1 Z a n,β,h E exp n k=1 (h + βω k )δ k δ n , n ∈ N, β 0, h ∈ R, where (1.7) Z a n,β,h = EE e n k=1 (h+βω k )δ k δ n = E(Z n,β,h ) is the annealed partition function, and the annealed free energy is also defined by (1.8) F a (β, h) = lim n→∞ 1 n log Z a n,β,h , TOME 2 (2019) when the limit exists. The annealed features are then (1.9) h a c (β) = inf{h : F a (β, h) > 0}, ν a (β) = lim , when the limit exists (along the paper we may omit β to lighten the notation, when there is no ambiguity). A simple application of Jensen's inequality leads to the following comparison: (1.10) F (β, h) F a (β, h) for all h ∈ R, β 0, and consequently (1.11) h a c (β) h c (β). If the annealed and quenched critical points or exponents differ at a given value of β, then disorder is said to be relevant for this value of β.

The Harris criterion
There have been a lot of studies on this problem in the past few years in the case when disorder is given by a sequence of i.i.d. random variables with exponential moments (under this assumption the annealed model coincides with the homogeneous model after a suitable shift of h). All these works put the prediction known in the physics literature as the Harris criterion [Har74] on a firm mathematical ground, which in this context states that disorder should be irrelevant if α < 1/2 (at least for small values of β) and relevant if α > 1/2. Several approaches have been used: direct estimates such as fractional moment and second moment estimates [AS06, AZ09, BL16, BCP + 14, CdH13a, DGLT09, GT06, GTL10], martingale theory [Lac10], variational techniques [CdH13b], and more recently chaos expansions of the partition functions [CSZ17, CSZ16,CTT17]. The limiting case α = 1/2 has been the subject of a lot of controversies and has been fully answered only recently [BL16]. Finally, the full criterion for relevance (in the sense of critical point shift) reads (1.12) ∀ β > 0, h c (β) > h a c (β) ⇐⇒ n 1 L(n) −2 n 2(α−1) = ∞, that corresponds to the intersection of two independent copies of τ being recurrent.

Correlated disorder: state of the art
The study of pinning models in correlated disorder is more recent, see [Ber13,Ber14,BL12,BP15,Poi13b]. From a mathematical perspective, it is quite natural to try and understand how crucial the assumption of independence is for the basic properties of the polymer, and in particular for the validity of the relevance criterion. Also, in several instances, the sequence of inhomogeneities may present more or less strong correlations: let us mention for instance the sequence of nucleotides which play the role of the disorder sequence in DNA denaturation [JPS06]. The main idea is that the relevance criterion should be modified only if the correlations are strong enough. Note that with correlated disorder, even the annealed model may not be trivial. Mainly two types of correlated disorder have been considered until now: correlated Gaussian disorder [Ber13,BP15,Poi13b] and random environments with large attractive regions of sub-exponential decay, also referred to as infinite disorder [Ber14,BL12].

Scope of the paper
The disorder sequence we consider is based on another renewal sequenceτ , independent of τ , starting at the origin and whose law shall be denoted byP. More specifically, we assume that if the Markov chain visits the origin at time n then it is given a reward equal to one if n ∈τ , zero otherwise. We are therefore dealing with the following binary correlated disorder sequence: From now on, the inter-arrival laws of τ andτ satisfy (1.14) (1.15) µ := E(τ 1 ),μ :=Ê(τ 1 ), which may be finite or infinite. Note that these definitions ensure aperiodicity for both renewal processes. In principle, the constants c K and cK may also be replaced by slowly varying functions, which would allow inclusion of the special case α ∈ {0, 1} in the discussion, but we refrain from doing so for the sake of simplicity. Also, we write an equality in the definition ofK(n) to ensure log-convexity. This technical condition is actually only needed for proving Theorem 2.8 (see Lemmas 4.2 and 4.3), which we actually believe to hold when the equality sign is replaced by the equivalent sign in the definition ofK in (1.14). The definitions of the basic thermodynamical quantities is the same as in the previous section, except that P and E are replaced byP andÊ. The condition that n ∈ τ in the definition of the polymer measures above could be removed, leading to the free versions. The versions with this condition are called the pinned versions. It is a standard fact [Gia07, Remark 1.2] that this minor modification does not have any effect on the limiting free energies as defined in Propositions 2.1 and 2.7.
A first dichotomy arises: • Ifα < 1, then the quantity in front of β in the Hamiltonian is at most |τ ∩ {1, . . . n}|, which is of order nα = o(n), and therefore disorder has no effect on the quenched free energy, which reduces to the homogeneous free energy. However the annealed model is non-trivial, so we include this case for completeness. • Ifα > 1, then (i) we may replaceP by its stationary version, denoted byP s , under which the distribution of the increments (τ n+1 −τ n ) n∈N 0 is the same as inP, whereas that ofτ 0 becomes {P(τ 1 > n)/μ} n∈N 0 , see e.g. [Asm03, Chapter V, Corollary 3.6] (again, this does not affect the free energy, see Propositions 2.1 and 2.7); (ii) the correlation exponent of our environment iŝ α − 1, since for n > m, for some positive constant c. The latter can be deduced from the Renewal Theorem and the following renewal convergence estimate [Fre82, Lemma 4] Although our choice of disorder may seem at first quite specific, it is motivated by the following: • By tuning the value of the parameter exponentα, one finds a whole spectrum of correlation exponents ranging from non-summable correlations to summable correlations, according to whether the sum n 0 CovP s (δ 0 ,δ n ) is infinite or finite. According to (1.16), correlations are summable whenα > 2 and non-summable whenα < 2.
• Our disorder sequence is bounded, therefore the annealed free energy is always finite, in contrast to the case of Gaussian variables with non-summable correlations [Ber13]. • The probability of observing a long sequence of ones decays exponentially with length, which rules out the infinite disorder regime discussed in [Ber14]. • The renewal structure of the disorder sequence makes the study of the annealed model and decoupling inequalities more tractable.

Summary of our results
What we prove in this paper is the following: for the caseα > 2 (summable correlations), disorder is irrelevant if α < 1/2 (both in the sense of critical points and critical exponents, at least for small values of β) and relevant if α > 1/2 (in the sense of critical exponents), which extends the Harris criterion for independent disorder. For the caseα ∈ (1, 2) (non-summable correlations) all we are able to prove is that disorder is relevant when α > 1/α, a condition that we expect to be non-optimal. We discuss a list of predictions for disorder relevance in that case. Finally, in the caseα ∈ (0, 1) disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.

Outline
We present our results in Section 2. Section 2.1 is dedicated to the annealed model and Section 2.2 to the quenched model. The proofs for the annealed model are in Section 3, and for the quenched model in Section 4. Results on renewal theory and homogeneous pinning are summarized in the appendix.

Results
The intersection set of τ andτ , which we denote by will play a fundamental role in the remainder of the paper. Note that it is itself a renewal starting at τ 0 = 0. We denote its law by P and write

Results on the annealed model
We begin with the existence of the annealed free energy.
Proposition 2.1. -For all β 0 and h ∈ R, the annealed free energy exists and it is finite and non-negative. The result still holds, without changing the value of the free energy, whenμ < ∞ andP is replaced byP s .
The following basic properties of the annealed free energy are standard: the function (β, h) → F a (β, h) is convex, continuous, and non-decreasing in both variables.
2.1.1. An auxiliary function: the number of intersection points Before stating further results, we need to introduce an auxiliary function which will help us to characterize the annealed critical point. For h 0, denote by P h the probability of the transient renewal process with τ 0 = 0 and inter-arrival law We denote the corresponding expectation by E h . The expected number of points in the renewal set τ (including 0) under the law P h ×P is denoted by Note that (2.6) The function I is finite and infinitely differentiable in (−∞, 0). It is also continuous in (−∞, 0], increasing, and strictly convex. Its range is [1, I(0)] with I(0) = EÊ(|τ ∩τ |), which may be finite or infinite. It follows from Proposition A.1 that Our next result provides an expression for the annealed critical curve involving the function I.
is nonnegative. Therefore, using Proposition A.3, we see that By the properties of I, The next two propositions provide the scaling behaviour of the annealed critical curve close to β 0 .
Our next result is about the order of the annealed phase transition.
Therefore, the annealed critical exponent remains unchanged compared to the homogeneous case ifα > 1, but is changed for large values of β whenα < 1 and α < 1.

Results on the quenched model
We start with the existence of the quenched free energy.
We are able to prove the following smoothing inequality.
We also prove the following result on disorder irrelevance.
Theorem 2.9. -Ifα > 2 and α < 1/2, then disorder is irrelevant for β small enough, meaning that h c (β) = h a c (β) and To the best of our knowledge, such a result on disorder irrelevance (in both critical points and exponents) has not yet been proven for other instances of correlated disorder, e.g. Gaussian disorder with summable correlations.
Whenμ is infinite, the issue of critical point shift is settled thanks to Propositions 2.2 and 2.7, which tell us that h c (β) = 0 for all β 0. Thus we get that h a c (β) = h c (β) when β − log (0) and h a c (β) < h c (β) when β > − log (0). The next proposition gives a condition under which h a c (β) < h c (β) for large β whenμ is finite.
Finally, our results on the issue of disorder relevance are summed up in Figure 2.2.

Discussion
We collect here remarks about our results.
(1) Note that when β 0 > 0, that is, when EÊ(|τ ∩τ |) < ∞, then for small β, the annealed critical exponent is the same as in the case when the renewalτ is absent. The reason behind this is that the reward β given at each intersection point in τ ∩τ is too weak forτ to contribute to the free energy.
(2) According to the Weinrib-Halperin criterion [WH83], which aims to generalize the Harris criterion, disorder should be relevant if ν < 2 ξ∧1 (at least for small disorder) and irrelevant if ν > 2 ξ∧1 , where ν is the critical exponent of the pure (homogeneous) system and ξ is the correlation exponent of the environment. The application of this criterion to pinning models was introduced and discussed in [Ber13]. In our case, ν = (1/α) ∨ 1, ξ =α − 1 (assuming thatα > 1), and the Harris criterion should not be changed if ξ > 1, i.e.α > 2, which is confirmed by Theorems 2.8 and 2.9. If α ∈ (1, 2), the criterion predicts that disorder is relevant (resp. irrelevant) if α >α −1 2 (resp. α <α −1 2 ). However, there is no clear evidence that this criterion gives the right prediction outside the Gaussian regime and it has actually been disproved in several examples [Ber13,Ber14].
(3) The recent work of Caravenna, Sun and Zygouras [CSZ17] has opened a new perspective on the issue of disorder relevance. Their work examines conditions under which we may find a weak-coupling limit of quenched partition functions, with randomness surviving in the limit. More precisely, they determine conditions under which there exist sequences of parameters in the Hamiltonian (the coupling constants h n , β n in our case) that converge to zero as the size of the system goes to infinity, and such that the properly rescaled quenched partition function converges in distribution to a random limit, which is obtained in the form of a Wiener chaos expansion. In several instances, including the one of the pinning model in an i.i.d. environment, it was shown that these conditions coincide with those of disorder relevance. Applying this approach to our model leads to the following conjecture.
Conjecture 2.11. -Disorder is relevant for all β > 0 (in the sense of critical point shift) if .
This problem will be examined in a future work. The reason for the term 1/α in place of the usual 1/2 whenα ∈ (1, 2), is that the partial sums of our disorder sequence is in the domain of attraction of anα-stable law. More specifically: (2.21) 1 n 1/α n k=1 (δ k − 1/μ) −→α-stable law, as n → ∞,α ∈ (1, 2). Therefore we expect that white noise is replaced by a Levy noise in the weak-coupling limit of the quenched partition function. Note that (2.19) and (2.20) coincide with the case of i.i.d. disorder whenα > 2, that is the summable correlation scenario. Finally, another reason to believe in this conjecture is that the chaos expansion approach gives the right prediction for a pinning model in an i.i.d. γ-stable environment (1 < γ < 2), which has been studied recently by Lacoin and Sohier [LS17]. There, it has been proved that disorder is relevant (resp. irrelevant) if α > 1 − 1/γ (resp. α < 1 − 1/γ), which is to be compared to our conjecture.
(4) The picture that has emerged for the moment regarding disorder relevance for this model can be summed up in the exponent diagram, see Figure 2.2.
• The blue area is where we have proven relevance for small β.
• In the regionα < 1 we have relevance in the blue area because the quenched critical curve is trivially 0 while the annealed is strictly negative. • In the blue region withα > 1, we have relevance due to smoothing, see Theorem 2.8. We do not know yet whether the critical points differ but we conjecture that they do (see Conjecture 2.11 above). • In the yellow triangle we have irrelevance because there both critical curves are 0 for small β, and the critical exponents agree. • In the yellow part withα > 2, we have irrelevance due to Theorem 2.9.
(5) Finally, let us mention the recent work of Alexander and Berger [AB18] who also consider a pinning model with disorder built out of a renewal sequence. Even if they may look similar, the model studied in [AB18] and the one considered in this paper are actually different in spirit. Indeed, in [AB18] all the interactions up to the nth renewal point of the disorder renewal (denoted here byτ n ) are taken into account, and the only parameter is the inverse temperature β (no pinning strength h). As a consequence, the results obtained therein are also quite different as for instance, the critical line deciding disorder relevance is at α +α = 1. However, we do not exclude that the two models are related. For instance, the line α +α = 1 also appears in Remark 2.3 above and, incidentally, in Proposition 2.6 (see also Figure 2.2).

Proof of the annealed results
The main idea is that the annealed model can be viewed as a homogeneous pinning model for the intersection renewal τ = τ ∩τ after the law of τ has been tilted.

Existence of the free energy
Proof of Proposition 2.1. -We use standard techniques, see the proof of Lemma 3.5 in [Gia11]. Let us first introduce the fully-pinned annealed partition function We shall write Z a,c n,β,h (A) when the expectation above is restricted to the event A and use the same convention for other versions of the partition function appearing in the proof. By super-additivity, the sequence {n −1 log Z a,c n,h,β } n∈N converges to the limit From the bounds From (3.2) we know that Z a,c r,β,h e rF a,c (β,h) . Moreover, Thus, By Proposition A.3, the last sum increases polynomially in n. Combining this inequality with Z a,c n,β,h Z a n,β,h , we get that the free energy F a (β, h) exists and equals F a,c (β, h).
We now prove the second part of the result, namely that the limit forP s is the same as forP. Suppose thatμ < ∞, and define Z a,s n = EÊ s exp n k=1 (h + βδ k )δ k δ n (we temporarily remove β and h, for conciseness). By restricting the expectation to the event {0 ∈τ }, we get on the one hand Z a,s n Z a n /μ. On the other hand, by decomposing on the value of τ 1 , we obtain To go from the first to second line, we used the Markov property at τ 1 and the fact that δ kδk = 0 for all k < τ 1 . Combining (3.7), (3.6), and using that hence the result.

Annealed critical curve
This subsection is organized as follows: we start with Lemma 3.1 below, which we use to prove Proposition 2.2. From Lemma 3.2 we get Proposition 2.5 and Lemma 3.3, which in turn yields Proposition 2.4. Proof of Lemma 3.1. -Let h 0. We know from the proof of Proposition 2.1 that F a (β, h) is the limiting free energy of the fully-pinned partition function in (3.1), which we may rewrite, using (2.4) and (2.2), as which is the partition function of the homogenous pinning model with reward β for the renewal τ under the law P h ×P. It now follows from a standard fact about homogeneous pinning models (see [Gia07, Section 1.2.2, Equation (1.26)]) that its critical point (as β varies and h is fixed) is at Proof of Proposition 2.2. -We distinguish two cases. Suppose first that β Theorem 2.7], since τ is recurrent. Thus, h a c (β) = 0 in this case. Suppose now that β > β 0 . If h 0, F a (β, h) > 0 for the same reason as above, so we restrict to h < 0. Then, by Lemma 3.1, (1 − e −β ) −1 (recall (2.7) and the lines above). Since β < β 0 , we have hence the result.
Proof of Proposition 2.5. -Since β 0 > 0 we have by Proposition 2.2 Moreover, for some positive constant c > 0, as h → 0 − . The equivalence above follows from Lemma A.8. We get the final result by noting that h a c (β 0 ) = 0 and combining (3.15) and (3.16).
where X α is an α-stable random variable totally skewed to the right, with scale parameter σ > 0 depending on the distribution of τ 1 and shift parameter 0 (see relation (A.3) in the appendix for a reminder of these terms).
Since τ k k, we may take the expectation in the line above and write and we may conclude the proof with Lemma A.5.
and with the same argument as in (i), We may conclude thanks to Lemma A.5.
where c is a positive constant (note that in the first caseμ = ∞ and the left-hand side is simply I(h)). Moreover, in the caseα > α ∧ 1 + 1, the constant c satisfies c > (1 −μ −1 )/2.
Proof of Lemma 3.3. -Recall (2.6). For h < 0, we may write Then, in the first three cases, the result follows from Lemma 3.2 and the standard Tauberian arguments recalled in Lemma A.8 (iii). In the fourth case, we have Note that the set σ := {k ∈ N 0 : τ k ∈τ } defines a renewal process. Call σ 1 its first positive point. This has meanμ because P ×P(k ∈ σ) → 1/μ as k → ∞. Let us call v ∈ (0, ∞] its variance. We now use Problem 19 in Chapter XIII of [Fel68], which needs a fix: we let the reader check that in equations (12.1) and (12.2) therein, u 0 should be removed and the sums should start at n = 0. According to this, we have

The annealed critical exponent
Recall the definition of P h in (2.4). When β > β 0 = − log (0), it will be useful for the proof of Proposition 2.6, which follows, to introduce the inter-arrival law We denote by E β the expectation with respect to this law (note that for β = 0, it coincides with the law of τ under P ×P). It is a by-product of the proof of Lemma 3.1 that h a c (β) is chosen such that the sequence ( K β (n)) n∈N sums up to 1, and thus defines a recurrent renewal.
Remark 3.4. -Note that if F β is the free energy of the homopolymer with interarrival law distribution n → P h a c (β) ×P((τ ∩τ ) 1 = n), then the homopolymer with inter-arrival law distribution K β and pinning reward h 0 has a free energy equal to F β (β + h).
Let us start with a uniform estimate on the mass renewal function under P h ×P, that will be used in the proof of Proposition 2.6 below.
Proof of Lemma 3.5. -First, the probability in the display equals P h (n ∈ τ ) P(n ∈τ ). As a lower bound for P h (n ∈ τ ), we get while for an upper bound we let h 0 = sup J < 0 and use the fact that P(τ k = n) k c K(n) for all n, k 1 and some constant c > 0 (see [Gia07, Lemma A.5]) to get for all h ∈ J. It is crucial here that the compact set J does not include 0 so that h 0 < 0. Combining these estimates with Proposition A.3, we get the result.
Proof of Proposition 2.6. -We consider the two possible cases for β.
Then h a c (β) < 0 by Proposition 2.2. Lower bound. -We bound the partition function from below, as follows: where in the last equality we use (3.26). By Lemma 3.5 applied to the set J = {h a c (β)}, we know that the mass renewal function n → P h a c (β) ×P(n ∈ τ ) satisfies (3.27). The lower bound then follows if one recalls Remark 3.4 and uses Lemma A.6, where the singleton {β} and the renewal n → P h a c (β) ×P(τ ∩τ ) 1 = n) play the role of the compact set I and the renewal K γ therein.

Existence of the free energy
Proof of Proposition 2.7. -When β = 0, the model reduces to the homogeneous pinning model, for which we know that the free energy F (0, h) exists. Therefore we assume that β > 0, and consider two cases.
We now prove that F s (β, h) = F (β, h). Assume in the rest of the proof thatτ is distributed according toP s . In the following we shall temporarily write Z n with a superscript indicating in which environment we are considering the partition function. By imposing the first step of τ to be equal toτ 0 , we get (4.8) In the other direction, we get, since |τ ∩ (0,τ 0 ]| τ 0 , (4.9) whereŽ n is the partition function obtained by replacing P by a slightly modified lawP in which the distribution of the first inter-arrival τ 1 is the one of the overshoot of τ with respect toτ 0 , namely (4.10)P(τ 1 = ) = P(inf{n τ 0 : n ∈ τ } =τ 0 + ), 0.
Note that this distribution depends on the random variableτ 0 . By decomposing on this first inter-arrival, we get where in the last inequality we have bounded the probabilities by one and used that , similarly to (4.8). Combining (4.9) and (4.11), and using that n k=1 K(k) −1 is only polynomially increasing in n, we may now compare the almost-sure limits of (1/n) log Zτ n+τ 0 and (1/n) log Zτ −τ 0 n , and obtain that F s (β, h) F (β, h). This completes the proof.

Smoothing inequality
This section is devoted to the proof of Theorem 2.8. Our proof is inspired by the original work of Giacomin and Toninelli in the i.i.d. disorder set-up [GT06] and is based on a localization strategy, which for the polymer consists in hitting favorable regions of the environment, where disorder is tilted. The cost of finding such regions is given by the entropy estimate in Lemma 4.1. Inspired by Caravenna and den Hollander [CdH13a], we also compare the free energy for a tilted disorder with that for a shifted disorder, see Lemma 4.3.
Let us start by considering the family of tilted disorder measures defined by (4.12) dP n,θ Note that this is nothing other than a pinning measure for the disorder renewal, and the normalizing constant in (4.12) is just the partition function of the homopolymer (β = 0) with pinning strength θ and underlying renewalτ . The corresponding relative entropy rate is defined as The proof of Lemma 4.1 below shows that this limit exists. Furthermore, it is nonnegative as the limit of non-negative real numbers. For the proof of Theorem 2.8 we will need three lemmas, which we now state and prove.   We may prove that this is actually an equivalence, but we will not need that fact. Proof of Lemma 4.2. -Pick k ∈ {1, . . . , n−1} and η ∈ {0, 1} [n] (the case k = n is trivial since both sides of (4.17) equal one). Writing the definition of the conditional probability in (4.17), we see that proving the claimed monotonicity is equivalent to proving that the function with η η and define a , b for η similarly as for η above. Necessarily 0 a a < k < b b n and we would like to have that This actually follows from the log-convexity ofK. Indeed, which, by log-convexity and since b − a k − a , is greater than which in turn, since b − a b − k, is greater than the right-hand side of (4.20).
(1) We say that a sequence of positive real numbers {u n } n∈N is log-convex if u 2 n u n−1 u n+1 for all n 2. It is equivalent to that un+m  Proof. -Define The idea is borrowed from [CdH13a] and consists of deriving a differential inequality for F n involving its partial derivatives with respect to h and θ. As can easily be checked, where P n,β,h is the Gibbs law as defined in (1.2), and For the random configurationδ ∈ {0, 1} [n] , we consider the following functions (4.28)δ →δ k −Ê n,θ (δ k ) (∀ k ∈ [n]) andδ → log Z n,β,h |δ k =y (∀ y 0).
Note that the denominators in (4.32) are nonzero and that C n,θ is positive, because we have previously removed the case k = n.
To estimate the integrals in the second line of (4.32), we will show that each f k is almost constant in y. Indeed, let us first write its derivative as (4.34) The first line shows that f k is non-decreasing in y while the second line shows that e −βy f k (y) is non-increasing in y. This gives in particular that f k (y 1 ) e −β f k (y 2 ) for all y 1 , y 2 ∈ [0, 1]. Looking back at (4.32), we obtain E n,θ 1 = e −βÊ n,θ E n,β,h (δ k ). (4.35) Taking into account (4.26), (4.30), (4.32) and (4.35) we obtain for β ∈ (0, 1) where c = (1/e) inf θ θ 0 inf n 1 C n,θ is positive for θ 0 small enough (we shall prove this point later) and the term 1/n above comes from the fact that the term k = n appears in (4.26) but not in (4.32). Therefore we obtain Thus the function g(t) := F n (β, h + cβθ(1 − t); θt), having non-negative derivative, is non-decreasing in [0, 1]. Therefore, g(0) g(1), which means that (4.38) F n (β, h + cβθ; 0) F n (β, h; θ) + cβθ n .
Therefore, we conclude the desired result by taking the superior limit as n → ∞.
Step 1. Lower bound on F (β, h). -Let us cut the system into blocks of size m ∈ N, namely A block is declared to be good if on this block the environment is favorable for the polymer. More precisely, pick a ∈ (0, 1) (for the moment its precise value is irrelevant) and define (recall (4.2)) (4.43) G = j 0 : Z jm,(j+1)m,β,h exp{aÊ θ (τ 1 ) −1 F (β, h; θ)m} , θ > 0, whereP θ is the law of a renewal with inter-arrival distributionP θ (τ 1 = n) = exp(θ − F (θ)n)K(n), for n 1, and F (β, h; θ) is defined in (4.24). The factorÊ θ (τ 1 ) −1 in the exponential is essentially harmless and the reason for its presence will appear clearer at the end of the proof. We now consider the blocks for which both endpoints are inτ , i.e., Then, denote by {σ k } k∈N the elements of J ∩ G, which form a renewal sequence. Our first task is to relate F (β, h) to the free energy associated to partition functions whose endpoints are in J ∩G. By Kingman's subadditive ergodic theorem there exists a non-negative number F(β, h) such that Step 2. Lower bound on the right-hand side of (4.48).
-We now apply the following strategy: the polymer makes a large jump until σ 1 m and visits the atypical block [σ 1 m, (σ 1 + 1)m]. Using (4.43), we obtain the following lower bound: and looking back at (4.48), we are left with bounding from below the quantity By Jensen's inequality,Ê(log σ 1 ) logÊ(σ 1 ). To boundÊ(σ 1 ), enumerate J as an increasing sequence (j k ) k∈N and define R as the unique integer for which σ 1 = j R . By the Markov property, the random variable R has a geometric distribution with probability of success (for the sake of clarity, we will establish the latter at the end of the proof), from which we obtain Therefore, Step 3. Conclusion of the proof. -Using (4.48) at h = h c (β), (4.55), (4.13) and letting m ↑ ∞, then a ↑ 1, we get One can check that (1.14) impliesK(n) 2 K (n−1)K(n+1) for n 2, so {K(n)} n∈N is log-convex. Therefore, we may apply Lemma 4.3, along with Lemma 4.1 and the fact thatÊ θ (τ 1 ) →μ as θ → 0, to get the desired estimate.
Step 4. Proof of (4.53). -Using the fact that for any two probability measures µ, ν on the same space and any event E with ν(E) > 0, it holds , it is actually enough to prove that For the proof of (4.58), the idea is to compute this limit whenP m,θ is replaced bŷ P θ , and then relate the two measures. The crucial observation is that for all bounded measurable functions φ, Using the same arguments as in Proposition 2.7, we get Moreover, Indeed, by (4.59), and (4.61) follows by considering the superior limit and using the Renewal Theorem. Finally, by using (4.59) one more time, we get that which goes to 1 as m → ∞ by virtue of (4.60) and (4.61), and since a < 1. This completes the proof.
As α → ∞, the first sum goes toK(1), while lim α→∞ c α log c α = 0. In the second sum, the n = 2 term converges to zero as α → ∞, and the same is true for the rest of the sum because it is bounded by (1 + α) ∞ 2 x −1−α log x dx < ∞.

Irrelevance. Proof of Theorem 2.9
The proof of Theorem 2.9, which can be found at the end of this section, essentially relies on Lemma 4.4 below, which is a control on the second moment of the partition function at the annealed critical point. Second moment methods and replica arguments have been used in the i.i.d. case in [Ton08]. The extra difficulty in our context is to deal with correlations, which we tackle by means of a decoupling inequality, see (4.95).
Lemma 4.4. -Supposeα > 2 and α < 1/2. Then, for β small enough, Proof of Lemma 4.4. -The proof is split into several steps. In Step 0, we change the partition function to a slightly modified version, which turns out to be more convenient in our context. In Step 1, we provide alternative expressions for the first and second moments of the partition function, using a cluster (or Mayer) expansion. In Step 2, we prove the decoupling inequality in (4.95), which we use in Step 3 to bound the second moment from above, uniformly in the size of the polymer. In Step 4 we prove that a simplified version of this upper bound is finite. The idea is to use the correlation decay ofτ (α > 2) to reduce the problem to the usual second moment control for pinning models with i.i.d. disorder, which in turn makes use of the transient nature of the overlap of two copies of τ (0 < α < 1 2 ), see [Gia11] and references therein. In Step 5, we conclude the main line of proof. Finally, Step 6 proves a technical point used in Step 4. Note that during this proof we will work only with free partition functions, i.e., δ n is removed from the definitions in (1.2), (1.3), (1.6) and (1.7).
Step 0. -During this proof we shall use the convention introduced below (3.1). We introduce, for h 0, where P h is defined in (2.4), and note thatZ n,β,h (n ∈ τ ) = Z n,β,h (n ∈ τ ), as it was already observed in (3.9). By decomposing according to the last renewal before n, we get (4.87) Therefore, it is enough to prove that Step 1. -We first rewrite the first and second moments of the modified partition functions. Namely, for h < 0, Let us first prove (4.89). Since δ kδk is {0, 1}-valued, we may write which gives (4.89) after interchanging sum and expectation. To get (4.90), we use the so-called replica trick and obtain (4.93)Ê(Z 2 n,β,h ) =ÊE ⊗2 h (e β 1 k nδ k (δ k +δ k ) ), where τ is an independent copy of τ . Then, and again the result follows by interchanging sum and expectation.
Step Definition of the gaps. -When one of I, J is empty, we let g(I, J) = 0. Then, the last product in (4.95) is 1 and the inequality is obviously true (note thatÛ (∅) = 1). When I and J are both non-empty, we give a recursive definition of the gaps using a backward exploration, which will prove to be useful later in the proof of (4.95). Therefore, we first need to define the last gap of two sets I and J that are both non-empty, which we denote by gap(I, J). Let us write (4.97) I = {i 1 , i 2 , . . . , i k } with i 1 < i 2 < · · · < i k , J = {j 1 , j 2 , . . . , j } with j 1 < j 2 < · · · < j .
If i k = j then w.l.o.g. we may assume that i k < j , in which case we define Indeed, if i k < j (using the notations above), then, Therefore,  r(gap(I, J)).
Step 3. -Recall (4.89). From Lemma 4.5 below, we have where the gaps (∆ m ) 1 m g(I,J) are associated to two sets I and J drawn independently from P.
Step 4. -This is an intermediate step to control the right-hand side of (4.109). If the sets {m 1 : ∆ m = i} therein were all replaced by {m 1 : ∆ m = 0}, it would be enough to control the following quantity: (4.110) Z := E C #{m 1 : ∆m=0} with C := e 2 i 0 r(i) .
We prove in this step that Z is finite if β is small (recall that P depends on z = z(β)). Note that C is finite becauseα > 2 and r(i) = O(i 1−α ). Indeed, for i 0, by (1.17). Then, observe that where τ refers to an independent copy of τ and the X i 's are independent Bernoulli random variables with parameter 1/C (it is clear from (4.110) that C > 1). Since the δ i X i 's and the δ j X j 's are {0, 1}-valued, we may write (4.115) Integrating over X, we get: Using that we obtain that (4.118) where (recall (4.91)) (4.119)β = log(1 + Cz 2 ) ∼ Cz 2 ∼ Cβ 2 as β 0.
But this is satisfied when h = h a c (β) and β is small enough since, as β 0, exp(β) converges to 1 and which is strictly less than 1 because α < 1/2 (see Proposition A.4). Note that the convergence in (4.124) does not seem to follow from simple arguments since Portmanteau's Theorem does not apply and Fatou's Lemma would go in the opposite direction. Therefore, we give an argument at the end of the proof, in Step 6, in order not to disrupt the main line of proof.
-We now prove that the right-hand side of (4.109) is finite using that Z is finite and a stochastic domination argument. For I, J finite subsets of N define Y i = #{m 1 : ∆ m = i} for all i ∈ N 0 . We claim that if I, J are independent, each with law P, then one can find for all i 1 a pair of random variables Y We will prove (4.125) at the end of this step. Apply Hölder's inequality with (4.126) Step 6. -It remains to prove the convergence in (4.124). Let us use as a shorthand notation: We want to prove that which we do by means of the Fourier series (a similar convergence problem was treated in [Poi13a] with the same technique). By the Renewal Equation, it is actually enough to prove that that is, convergence of the series which may be seen as the L 2 -norm of some function. More precisely, if we define where Re( · ) is the real part of a complex number. Equation (4.134) can be recovered from equation (8) in [Spi13,Chapter II.9]. Note that the integral in the right-hand side of (4.134) is real and even in n ∈ Z, since then the complex exponential may be replaced by a cosine. Moreover, (4.135) P β ( τ 1 = n) e β+h a c P(τ 1 = n)P(n ∈τ ) = 1 µ e β+h a c P(τ 1 = n)(1 + o(1)), so E β ( τ 1 ) = ∞ and the first term in the right-hand side of (4.134) is zero. Therefore, the sum in (4.132) is related to the L 2 norm of 2 Re and we only need to show convergence of the L 2 -norms of Re[(1 − ϕ β ( · )) −1 ] to that of Re[(1 − ϕ 0 ( · )) −1 ] as β → 0. We now observe that is such that EÊ(Ψ β ) = 1. Since h a c (β) 0, we get by the Dominated Convergence Theorem that lim β→0 ϕ β (t) = ϕ 0 (t). To show the convergence of the L 2 norm of Re[(1 − ϕ β ( · )) −1 ] to the one of Re[(1 − ϕ 0 ( · )) −1 ], we show domination by a square integrable function in [−π, π] for all β ∈ [0, β 0 ] where β 0 > 0 is fixed. Note that First we treat values of t close zero. Since 1 − cos(t τ 1 ) 0 a.s., we may write for β ∈ [0, β 0 ]: (4.140) Note that the minimum in the line above is positive as a consequence of the Renewal Theorem (α > 1). To go from the second to the third line, we restrict the expectation to the event {τ 1 = n}, on which |τ ∩ (0, n]| = 1, and use the monotonicity of h a c (β). The last step is a standard Riemann sum approximation. Thus there is t 0 ∈ (0, π] so that for |t| t 0 and β ∈ [0, β 0 ] it holds Re(1 − ϕ β (t)) c |t| α for some constant c > 0. The function |t| −α is in L 2 [−π, π], since α < 1/2.
The last quantity does not depend on t and is positive. This completes the proof.
Proof of Theorem 2.9. -The proof can be done by following the same steps as in the proof of Theorem 4.5 in [Gia11]. The main ingredient is the fact that the sequence (X n ) n∈N := (Z n,β,h a c ) n∈N is uniformly integrable, which is proven in Lemma 4.4. The fact that our disorder is correlated does not necessitate any change here. However, it is useful that the disorder is bounded. The conclusion of Lemma 4.6 in [Gia11], needed in the proof, still holds because (X n ) n∈N satisfiesÊ(X n ) = E h a cÊ (e β n k=1 δ kδk ) 1, and this is enough for the proof of the theorem.
where P β has been defined in (3.26).

Appendix. Results on renewals and homogeneous pinning
We collect here a few results on renewal processes and the homogeneous pinning model. P(n ∈ τ ) ∼ P(τ 1 = n) P(τ 1 = ∞) 2 as n → ∞.
Before stating the next lemma, we recall that an α-stable random variable X has three parameters κ ∈ [−1, 1], σ 0 and m ∈ R (skewness, scale and shift, respectively) appearing in its characteristic function: see Definition 1.1.6 in [ST94]. If κ = 1 (resp. −1), X is said to be totally skewed to the right (resp. left).
Lemma A.5. -If r > 0, then where X α is an α-stable random variable totally skewed to the right, with scale parameter σ > 0 depending on the distribution of τ 1 and shift parameter 0.
Recall that F γ is zero exactly in (−∞, γ] and positive elsewhere.
which we write as Now for any function f : N 0 → [0, ∞), we definef (z) = ∞ n=0 f (n)z n for all z ∈ [0, 1]. Then the equality (A.14) In particular, e −γ = 1 − (û γ (1)) −1 , so that Of interest to us is the behavior of Ψ γ close to 0, and thus of A γ close to 1. The following claim addresses the issue. To state it, we let m := ∞ n=1 L(n)/n α .
Proof of the claim. -By the bounds we have on u γ , it suffices to examine the behavior of Denote by q k the coefficient of z k in this power series. We have q k ∼ L(k) αk α , for α > 0, by Proposition 1.5.10 in [BGT89], while for α = 0, q k is slowly varying (Proposition 1.5.9.b in [BGT89]). Thus, if α ∈ (0, 1), is slowly varying if α = 1 and m = ∞, → m if m < ∞.
We continue with the proof of the lemma. The claim above and (A.16) give that there are constants 0 < C 5 < C 6 and a slowly varying function L 2 so that (A.25) for all x ∈ (0, log 2] and γ ∈ I. Let C > 0 be fixed. For α > 0, by Proposition 1.5.15 in [BGT89], there is a slowly varying functionL so that a solution x C (h) of x α∧1 L 2 (1/x) = (1 − e −h )/C is asymptotically equivalent to a constant multiple of h 1∨1/αL (1/h) as h → 0 + (L is the same for all C). If α = 0, call x C (h) the smallest solution of L 0 (1/x) = (1 − e −h )/C. Then x C (h) = 1/L −1 0 ((1 − e −h )/C) with L −1 0 an obviously defined "inverse" of L 0 . It is easy to see, by bounding q k from below by ∞ n=k+1 L(n)/n 1+ε for any ε > 0, that each x C goes to zero faster than any power of h. Since Ψ γ (x) is increasing in x, for each γ ∈ I the solution of (A.12) is between x C 6 (h) and x C 5 (h) (x C 6 (h) < x C 5 (h)), which finishes the proof of the lemma.
In the following two lemmas we denote by P θ the renewal such that P θ (τ 1 = n) = e θ−F (θ)n K(n), where θ 0. Note that P 0 coincides with P. We also define φ θ (t) = E θ (e itτ 1 ), for t ∈ R.
Proof. -Throughout the proof, c will be a constant uniform in θ and t which may change from line to line.