Disorder relevance at marginality and critical point shift

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www.imstat.org/aihp 2011, Vol. 47, No. 1, 148–175

DOI:10.1214/10-AIHP366

Disorder relevance at marginality and critical point shift

Giambattista Giacomin

^a

, Hubert Lacoin

^a

and Fabio Lucio Toninelli

^b

aUniversité Paris Diderot (Paris 7) and Laboratoire de Probabilités et Modèles Aléatoires (CNRS), U.F.R. Mathématiques, Case 7012 (site Chevaleret) 75205 Paris Cedex 13, France. Emails:giacomin@math.jussieu.fr;lacoin@math.jussieu.fr

bCNRS and Laboratoire de Physique, ENS Lyon, 46 Allée d’Italie, 69364 Lyon, France. E-mail:fabio-lucio.toninelli@ens-lyon.fr Received 6 July 2009; revised 1 March 2010; accepted 9 March 2010

Abstract. Recently the renormalization group predictions on the effect of disorder on pinning models have been put on mathematical grounds. The picture is particularly complete if the disorder is relevantorirrelevantin the Harris criterion sense: the question addressed is whether quenched disorder leads to a critical behavior which is different from the one observed in the pure, i.e. annealed, system. The Harris criterion prediction is based on the sign of the specific heat exponent of the pure system, but it yields no prediction in the case of vanishing exponent. This case is calledmarginal, and the physical literature is divided on what one should observe for marginal disorder, notably there is no agreement on whether a small amount of disorder leads or not to a difference between the critical point of the quenched system and the one for the pure system. In [Comm. Pure Appl. Math.63 (2010) 233–265] we have proven that the two critical points differ at marginality of at least exp(−c/β⁴), wherec >0 andβ²is the disorder variance, forβ∈(0,1)and Gaussian IID disorder. The purpose of this paper is to improve such a result: we establish in particular that the exp(−c/β⁴)lower bound on the shift can be replaced by exp(−c(b)/β^b),c(b) >0 forb >2 (b=2 is the known upper bound and it is the result claimed in [J. Stat. Phys.66(1992) 1189–1213]), and we deal with very general distribution of the IID disorder variables. The proof relies on coarse graining estimates and on a fractional moment change of measure argument based on multi-body potential modifications of the law of the disorder.

Résumé. Récemment, les prédictions issues des méthodes de groupe de renormalisation concernant l’influence du désordre pour les modèles d’accrochage ont été rendus rigoureuses mathématiquement. La description du phénomène est particulièrement com- plète dans le cas où le désordre estpertinentounon-pertinentau sens du critère de Harris: on étudie si le désordre gelé engendre un comportement critique différent de celui que l’on observe pour le système pur, i.e. moyenné. Le critère de Harris se base sur le signe de l’exposant de la chaleur spécifique du système pur pour déterminer l’influence du désordre, mais ne prédit rien dans le cas où cet exposant vaut zéro. Ce cas est ditmarginalet il n’y a pas de consensus dans la littérature physique sur ce que l’on devrait observer pour le système désordonné marginal; en particulier, il y a une controverse pour déterminer si un désordre de faible amplitude engendre ou non un déplacement du point critique du système avec désordre gelé par rapport à celui du système pur. Dans [Comm. Pure Appl. Math.63(2010) 233–265], nous avons démontré que, dans le cas marginal, la différence entre les deux points critiques est au moins d’ordre exp(−c/β⁴), oùc >0 etβ²est la variance du désordre, pourβ∈(0,1)dans le cas d’un désordre gaussien IID. L’objectif de cet article est d’améliorer le résultat précédent: en particulier nous montrons que la borne inférieure exp(−c/β⁴)pour le déplacement du point critique peut être remplacée par exp(−c(b)/β^b),c(b) >0 pour toutb >2 (b=2 est la borne supérieure connue, et le résultat prédit dans [J. Stat. Phys.66(1992) 1189–1213]), et nous généralisons la preuve à des désordres IID très généraux. La démonstration s’appuie sur des estimées obtenues parcoarse graining, et sur l’estimation de moments non-entiers de la fonction de partition, en modifiant la loi du désordre en y appliquant un potentiel multicorps.

MSC:82B44; 60K35; 82B27; 60K37

Keywords:Disordered pinning models; Harris criterion; Marginal disorder; Many-body interactions

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1. Introduction

1.1. Relevant,irrelevant and marginal disorder

The renormalization group approach to disordered statistical mechanics systems introduces a very interesting view- point on the role of disorder and on whether or not the critical behavior of a quenched system coincides with the critical behavior of the correspondingpuresystem. The Harris criterion [17] is based on such an approach and it may be summarized in the following way: if the specific heat exponent of the pure system is negative, then asmallamount of disorder does not modify the critical properties of the pure system (irrelevant disorder regime), but if the specific heat exponent of the pure system is positive then even an arbitrarily small amount of disorder may lead to a quenched critical behavior different from the critical behavior of the pure system.

A class of disordered models on which such ideas have been applied by several authors is the one of pinning models (see, e.g., [8,11] and the extensive bibliography in [12,14]). The reason is in part due to the remarkable fact that pure pinning models are exactly solvable models for which, by tuning a parameter, one can explore all possible values of the specific heat exponent [10]. As a matter of fact, the validity of the Harris criterion for pinning models in the physical literature finds a rather general agreement. Moreover, for the pinning models the renormalization group approach goes beyond the critical properties and yields a prediction also on the location of the critical point.

Recently, the Harris criterion predictions for pinning models have been put on firm grounds in a series of papers [1,3,7,19] and some of these rigorous results go even beyond the predictions. Notably in [15] it has been shown that disorder has a smoothing effect in this class of models (a fact that is not a consequence of the Harris criterion and that did not find unanimous agreement in the previous physical literature).

However, a substantial amount of the literature on disordered pinning and Harris criterion revolves around a specific issue: what happens if the specific heat exponent is zero (i.e., atmarginality)? This is really a controversial issue in the physical literature, started by the disagreement in the conclusions of [11] and [8]. In a nutshell, the disagreement lies on the fact that the authors of [11] predict that disorder is irrelevant at marginality and, notably, that quenched and annealed critical points coincide at small disorder, while the authors of [8] claim that disorder is relevant for arbitrarily small disorder, leading to a critical point shift of the order of exp(−cβ⁻²)(c >0) forβ0 (β² is the disorder variance).

Recently we have been able to prove that, at marginality, there is a shift of the critical point induced by the presence of disorder [14], at least for Gaussian disorder. We have actually proven that the shift is at least exp(−cβ⁻⁴). The purpose of the present work is to go beyond [14] in three aspects:

(1) We want to deal with rather general disorder variables: we are going to assume only that the exponential moments are finite.

(2) We are going to improve the bound exp(−cβ⁻^b),b=4, on the critical point shift, tob=2+ε(ε >0 arbitrarily small, andc=c(b)).

(3) We will prove our results for a wide class of pinning models. Pinning models are based on discrete renewal processes, characterized by an inter-arrival distribution which has power-law decay (the exponent in the power law parametrizes the model and varying such parameter one explores the different types of critical behaviors we mentioned before). The general pinning model is obtained by relaxing the power law decay to regularly varying decay, that is (in particular) we allowlogarithmic correctionto power-law decay. This, in a sense, allows zooming into the marginal case and makes clearer the interplay between the underlying renewal and the disorder variables.

1.2. The framework and some basic facts

In mathematical terms, disordered pinning models are one-dimensional Gibbs measures with random one-body potentials and reference measure given by the law of a renewal process. Namely, pinning models are built starting from a (non-delayed, discrete) renewal processτ = {τ_n}n=0,1,..., that is a sequence of random variables such thatτ₀=0 and {τj+1−τj}j=0,1,... are independent and identically distributed with common law (calledinter-arrival distribution) concentrated on N:= {1,2, . . .}(the law ofτ is denoted byP): we will actually assume that such a distribution is regularly varying of exponent 1+α, i.e.

K(n):=P(τ1=n)=L(n)

n¹⁺^α forn=1,2, . . . , (1.1)

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where α≥0 and L(·) is a slowly varying function, that is L:(0,∞)→(0,∞) is measurable and it satisfies limx→∞L(cx)/L(x)=1 for everyc >0. There is actually no loss of generality in assuming L(·)smooth and we will do so (we refer to [4] for properties of slowly varying functions).

Remark 1.1. Examples of slowly varying functions includelogarithmic slowly varying functions (this is probably not a standard terminology,but it will come handy),that is the positive measurable functions that behave likea(log(x))^b asx→ ∞,witha>0and b∈R.These functions are just a particular class of slowly varying functions,but it is already rich enough to appreciate the results we are going to present.Moreover,we will say thatL(·)is trivial if limx→∞L(x)=c∈(0,∞).The general statements about slowly varying function that we are going to use can be verified in an elementary way for logarithmic slowly varying functions;readers who feel uneasy with the general theory may safely focus on this restricted class.

Without loss of generality we assume that

n∈NK(n)=1 (actually, we have implicitly done so when we have introducedτ). This does not look at all like an innocuous assumption at first, because it means thatτ ispersistent, namelyτ_j<∞for everyj, while if

nK(n) <1 thenτ isterminating, that is|{j: τ_j<∞}|<∞a.s. It is however really a harmless assumption, as explained in detail in [12], Chapter 1, and recalled in the caption of Fig.1.

The disordered potentials are introduced by means of the IID sequence {ω_n}n=1,2,... of random variables (the charges) such that M(t):=E[exp(tω1)]<∞ for every t. Without loss of generality we may and do assume that E[ω1] =0 and var_P(ω1)=1.

The model we are going to focus on is defined by the sequence of probability measuresPN,ω,β,h=PN,ω, indexed byN∈N, defined by

dPN,ω

dP (τ ):= 1 Z_N,ωexp

_N

n=1

βωn+h−log M(β) δn

δN, (1.2)

whereβ≥0,h∈R,δ_nis the indicator function thatn=τ_j for somej andZ_N,ωis the partition function, that is the normalization constant. It is practical to look atτ as a random subset of{0} ∪N, so that, for example,δ_n=1n∈τ.

Fig. 1. A symmetric random walk trajectory with increments taking values in{−1,0,+1}is represented as a directed random walk. On thex-axis, thedefect line, there are quenched chargesωthat are collected by the walk when it hits the charge location. The energy of a trajectory just depends on the underlying renewal processτ. For the case in the figure,K(n):=P(τ₁=n)∼constn⁻^3/2forn→ ∞(e.g., [12], Appendix A.6). Moreover, the walk is recurrent, so

nK(n)=1. There is however another interpretation of the model: the charges may be thought of as sticking toS, not viewed this time as a directed walk. If the walk hits the origin at timen, the energy is incremented by(βωn+h−log M(β)). This interpretation is particularly interesting for a three-dimensional symmetric walk inZ³: the walk may be interpreted as a polymer ind=3, carrying charges on each monomer, and the monomers interact with a point in space (the origin) via a charge-dependent potential. Also in this caseK(n)∼constn^−3/2, but the walk is transient so that

nK(n) <1 (e.g., [12], Appendix A.6). It is rather easy to see that any model based on a terminating renewal with inter-arrival distributionK(·)can be mapped to a model based on the persistent renewal with inter-arrival distributionK(·)/

nK(n)at the expense of changinghtoh+log

nK(n). For much more detailed accounts on the (very many!) models that can be directly mapped to pinning models we refer to [10,12].

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Remark 1.2. We have chosen M(t) <∞ for every t only for ease of exposition. The results we present directly generalize to the case in whichM(t0)+M(−t₀) <∞for at₀>0.In this case it suffices to look at the system only for β∈ [0, t0).

Three comments on (1.2) are in order:

(1) We have introduced the model in a very general set-up which is, possibly, not too intuitive, but it allows a unified approach to a large class of models [10,12]. It may be useful at this stage to look at Fig.1 that illustrates the random walk pinning model.

(2) The presence of−log M(β)in the exponent is just a parametrization of the problem that comes particularly handy and it can be absorbed by redefiningh.

(3) The presence ofδ_N in the right-hand side means that we are looking only at trajectories that arepinnedat the endpoint of the system. This is just a boundary condition and we may as well remove δ_N for the purpose of the results that we are going to state, since it is well known, for example, that the free energy of this system is independent of the boundary condition (e.g., [12], Chapter 4). Nonetheless, at a technical level it is more practical to work with the system pinned at the endpoint.

The (Laplace) asymptotic behavior ofZ_N,ωshows a phase transition. In fact, if we define the free energy as

F(β, h):= lim

N→∞

1

NElogZN,ω, (1.3)

where the limit exists since the sequence{ElogZ_N,ω}N is super-additive (see, e.g., [12], Chapter 4, where it is also proven that F(β, h)coincides with theP(dω)-almost sure limit of(1/N )logZ_N,ω, so thatF(β, h)is effectively the quenchedfree energy), then it is easy to see thatF(β, h)≥0: in fact,

F(β, h)≥lim sup

N→∞

1 NElogE

exp

_N

n=1

βω_n+h−log M(β) δ_n

1τ1=N

= lim

N→∞

1 N

h−log M(β)

+logP(τ1=N )

=0. (1.4)

The transition we are after is captured by setting h_c(β):=sup

h: F(β, h)=0

=inf

h: F(β, h) >0

, (1.5)

where the equality is a direct consequence of the fact thatF(β,·)is non-decreasing (let us point out also that the free energy is a continuous function of both arguments, as it follows from standard convexity arguments). We have the bounds (see point (2) just below for the proof)

F

0, h−log M(β)

≤^F(β, h)≤^F(0, h), (1.6)

which directly imply

h_c(0)≤h_c(β)≤h_c(0)+log M(β). (1.7)

Two important observations are:

(1) The bounds in (1.6) are given in terms of F(0,·), that is the free energy of the non-disordered system, which can be solved analytically (e.g., [10,12]). In particular hc(0)=0 for everyα and every choice ofL(·)(in fact hc(0)= −log

nK(n) and we are assuming that τ is persistent). We will keep in our formulae hc(0)both because we think that it makes them more readable and because they happen to be true also ifτ were a terminating renewal.

(2) The upper bound in (1.6), that entails the lower bound in (1.7), follows directly from the standard annealed bound, that isElogZN,ω≤logEZN,ω, and by observing that theannealed partition functionEZN,ωcoincides

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with the partition function of the quenched model withβ=0, that is simply the non-disordered case (of course, the presence of the term−log M(β)in (1.2) finds here its motivation). The lower bound in (1.6), entailing the upper bound in (1.7), follows by a convexity argument too (see [12], Chapter 5).

Remark 1.3. It is rather easy(just take the derivative of the free energy with respect toh)to realize that the phase transition we have outlined in this model is a localization transition:whenh < h_c(β),forN large,the random setτ isalmost empty,while whenh > h_c(β)it is of size constN (in fact const=∂_hF(β, h)).Very sharp results have been obtained on this issue:we refer to[12],Chapters7and8,and references therein.

1.3. The Harris criterion

We can now make precise the Harris criterion predictions mentioned in Section1.1. As we have seen, in our case the pure (orannealed) model is just the non-disordered model, and the latter is exactly solvable, so that the critical behavior is fully understood, notably [12], Chapter 2,

alim0

logF(0, hc(0)+a)

loga =max

1,1

α

=:ν_pure. (1.8)

The specific heat exponent of the pure model (that is the critical exponent associated to 1/∂_h²F(0, h)) is computed analogously and it is equal to 2−ν_pure. Therefore the Harris criterion predictsdisorder relevanceforα >1/2 (2− ν_pure>0) and disorder irrelevanceforα <1/2 (2−ν_pure<0) at least forβ below a threshold, with α=1/2 as marginal case. So, what one expects is thatν_pure=ν_quenched(with obvious definition of the latter) ifα <1/2 forβnot too largeandν_pure=ν_quenchedifα >1/2 (for everyβ >0).

While a priori the Harris criterion attacks the issue of critical behavior, it turns out that a Harris-like approach in the pinning context [8,11] yields information also onh_c(β), namely thath_c(β)=h_c(0)ifα <1/2 andβagain not too large, whileh_c(β) > h_c(0)as soon asβ >0. For the sequel it is important to recall some aspects of the approaches in [8,11].

The main focus of [8,11] is on the caseα=1/2 and trivialL(·). In fact they focus on the interface wetting problem in two dimensions, that boils down to directed random walk pinning in(1+1)-dimensions. In this framework the conclusions of the two papers differ: [11] stands forh_c(β)=h_c(0)forβ small, while in [8] one finds an argument in favor of

h_c(β)−h_c(0)≈exp

−cβ⁻²

, (1.9)

asβ0 (withc >0 an explicit constant).

We will not go into the details of these arguments, but we wish to point out why, in these arguments,α=1/2 plays such a singular role:

(1) In the approach of [11] an expansion of the free energy to all orders in the variance of exp(βω1−log M(β)), that is(M(2β)/M²(β))−1^β∼⁰β², is performed. In particular (in the Gaussian case)

F

β, h_c(0)+a

=^F

0, hc(0)+a

−1 2

exp β²

−1

∂_aF

0, hc(0)+a2

+ · · · (1.10)

and, whenL(·)is trivial,∂aF(0, hc(0)+a)behaves like (a constant times)a⁽¹⁻^α)/αforα∈(0,1)(this is detailed, for example, in [13]) and like a constant forα >1. This suggests that the expansion (1.10) cannot work for α >1/2, because the second-order term, fora0, becomes larger than the first order term (a^max(1/α,1)). The borderline case isα=1/2, and trust in such an expansion forα=1/2 may follow from the fact thatβ can be chosen small. In conclusion, an argument along the lines of [11] predicts disorder relevance if and only ifα >1/2 (ifL(·)is trivial).

(2) The approach of [8] instead is based on the analysis of var_P(ZN,ω)at the pure critical pointhc(0). This directly leads to studying the random setτ :=τ ∩τ (it appears in the computation in a very natural way, we call it intersection renewal), withτan independent copy ofτ (note thatτ is still a renewal process): in physical terms,

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one is looking at thetwo-replica system. It turns out that, even if we have assumedτ persistent,τ may not be: in fact, ifL(·)is trivial, thenτ is persistent if and only ifα≥1/2 (see just below for a proof of this fact). And [8]

predicts disorder relevance if and only ifα≥1/2.

Some aspects of these two approaches were made rigorous mathematically: The expansion of the free energy (1.10) was proved to hold forα <1/2 in [16], and the second moment analysis of [8] was used to provedisorder irrelevance in [1,19], making it difficult to choose between the predictions.

We can actually find in the physical literature a number of authors standing for one or the other of the two predictions in the marginal caseα=1/2 (the reader can find a detailed review of the literature in [14]). But we would like to go a step farther and we point out that, by generalizing naively the approach in [8], one is tempted to conjec- ture disorder relevance (at arbitrarily smallβ) if and only if the intersection renewal is recurrent. Let us make this condition explicit: while one does not have direct access to the inter-arrival distribution ofτ, it is straightforward, by independence, to write the renewal function ofτ:

P(n∈τ )=P(n∈τ )². (1.11)

It is then sufficient to use the basic (and general) renewal process formula

nP(n∈τ )=(1−

nP(τ1=n))⁻¹to realize thatτis persistent if and only if

nP(n∈τ )= ∞. Since under our assumptions forα∈(0,1)[9], Theorem B, P(n∈τ )ⁿ^→∞∼ αsin(πα)

π

1

n¹⁻^αL(n), (1.12)

we easily see that the intersection renewalτ is persistent forα >1/2 and terminating ifα <1/2 (the caseα=0 can be treated too [4], andτ is terminating). In theα=1/2 case the argument we have just outlined yields

τ∩τis persistent ⇐⇒

n

1

nL(n)²= ∞. (1.13)

Roughly, this is telling us that the intersection renewalτ is persistent up to a slowly varying functionL(x)diverging slightly lessthan(logx)^1/2. In particular, as we have already pointed out, ifL(·)is trivial,τ is persistent.

Let us remark that the expansion (1.10) has been actually made rigorous in [16], but only under the assumption that the intersection renewalτ is terminating (that is,b>1/2 for logarithmic slowly varying functions).

Remark 1.4. In view of the argument we have just outlined,we introduce the increasing functionL:(0,∞)→(0,∞) defined as

L(x):=

x 0

1

(1+y)L(y)²dy, (1.14)

that is going to play a central role from now on.Let us point out that,by[4],Theorem1.5.9a,L(·)is a slowly varying function which has the property

xlim→∞L(x)L(x)²= +∞, (1.15)

which is a non-trivial statement whenL(·)does not diverge at infinity.Of course we are most interested in the fact that,whenα=1/2,L(x)diverges asx→ ∞if and only if the intersection renewalτ is recurrent(cf. (1.13)).For completeness we point out thatL(·)is a special type of slowly varying function(aden Haan function [4],Chapter3), but we will not exploit the further regularity properties stemming out of this observation.

1.4. Review of the rigorous results

Much mathematical work has been done on disordered pinning models recently. Let us start with a quick review of theα=1/2 case:

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• Ifα >1/2 disorder relevance is established. The positivity ofh_c(β)−h_c(0)(with precise asymptotic estimates asβ 0) is proven [2,7]. It has been also shown that disorder has a smoothing effect on the transition and the quenched free energy critical exponent differs from the annealed one [15].

• Ifα <1/2 disorder irrelevance is established, along with a number of sharp results saying in particular that, ifβ is not too large,h_c(β)=h_c(0)and that the free energy critical behavior coincides in the quenched and annealed framework [1,3,16,19].

In the caseα=1/2 results are less complete. Particularly relevant for the sequel are the next two results that we state as theorems. The first one is taken from [1] (see also [19]) and uses the auxiliary functiona0(·)defined by

a₀(β):=C₁LL⁻¹

C₂/β²L⁻¹

C₂/β²1/2

withC₁>0 andC₂>0, (1.16)

if limx→∞L(x)= ∞, anda₀(·)≡0, otherwise.

Theorem 1.5. Fixω₁∼N(0,1),α=1/2and choose a slowly varying functionL(·).Then there existsβ₀>0and a1>0such that for everyε >0there existC1andC2>0such that

1−ε≤^F(β, a)

F(0, a) ≤1 fora > a0(β), a≤a1andβ≤β0. (1.17)

This implies forβ≤β₀

h_c(β)−h_c(0)≤a₀(β). (1.18)

It is worth pointing out that Theorem1.5yields an upper bound matching (1.9) whenL(·)is trivial.

The next result addresses instead the lower bound onhc(β)−hc(0)and it is taken from [14].

Theorem 1.6. Fixω1∼N(0,1)andα=1/2.IfL(·)is trivial,thenhc(β)−hc(0) >0for everyβ >0and there existsC >0such that

h_c(β)−h_c(0)≥exp

−C/β⁴

(1.19) forβ≤1.

It should be pointed out that [14] has been worked out for trivialL(·), addressing thus precisely the controversial issue in the physical literature. The case of limx→∞L(x)=0 has been treated [2] (see [7] for a weaker result) wherehc(β)−hc(0) >0 has been established with an explicit but not optimal bound. We point out also that a result analogous to Theorem1.6has been proven for a hierarchical version of the pinning model [18] (see [14] for the case of the hierarchical model proposed in [8]).

The understanding of the marginal case is therefore still partial and the following problems are clearly open:

(1) What is really the behavior ofh_c(β)−h_c(0)in the marginal case? In particular, forL(·)trivial, is (1.9) correct?

(2) Going beyond the case of L(·)trivial: is the two-replica condition (1.13) equivalent to disorder relevance for smallβ?

(3) What about non-Gaussian disorder? It should be pointed out that a part of the literature focuses on Gaussian disorder, notably Theorem1.5, but this choice appears to have been made in order to have more concise proofs (for example, the results in [7] are given for very general disorder distribution). Theorem1.6instead exploits a technique that is more inherently Gaussian and generalizing the approach in [14] to non-Gaussian disorder is not straightforward.

As we explain in the next subsection, in this paper we will givealmostcomplete answers to questions (1)–(3). In addition we will prove a monotonicity result for the phase diagram of pinning model which holds in great generality.

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1.5. The main result

Our main result requires the existence ofε∈(0,1/2]such that L(x)=o

log(x)(1/2)−ε

asx→ ∞, (1.20)

that is limx→∞L(x)(log(x))⁻^(1/2)⁺^ε=0. Of course, ifL(·)vanishes at infinity, (1.20) holds withε=1/2. Going back to the slowly varying functionL(·), cf. Remark1.4, we note that, under assumption (1.20), we have

L(x)^x^→∞

_x

2

1

y(logy)¹⁻^2εdy= 1

2ε(logx)^2ε− 1

2ε(log 2)^2ε. (1.21)

Therefore, under assumption (1.20), we have that ifq > (2ε)⁻¹then

xlim→∞

L(x)

L(x)^2/(q⁻¹⁾ = ∞, (1.22)

which guarantees that givenq > (2ε)⁻¹(actually, in the sequelq∈N) andA >0, (β;q, A):=

inf

n∈N: L(n)/L(n)^2/(q⁻¹⁾≥Aβ⁻^2q/(q⁻¹⁾₋1

(1.23) is greater than 0 for everyβ >0.

Our main result is the following theorem.

Theorem 1.7. Let us assume thatα=1/2and that(1.20)holds for someε∈(0,1/2].For everyβ₀and every integer q > (2ε)⁻¹there existsA >0such that

hc(β)−hc(0)≥(β;q, A) >0 (1.24)

for everyβ≤β0.

The result may be more directly appreciated in the particular case ofL(·)of logarithmic type, cf. Remark1.1, withb<1/2, so that (1.20) holds withε <min((1/2)−b,1/2). By explicit integration we see thatL(x)∼(a²(1− 2b))⁻¹(log(x))¹⁻^2bso that

L(x)

L(x)^2/(q⁻¹⁾ ∼a⁻^2q/(q⁻¹⁾ (1−2b)

log(x)1−2bq(q−1)⁻¹

(1.25) and in this case

(β;q, A)^β∼⁰exp

−c(b, A, q)β⁻^b

, (1.26)

wherec(b, A, q):=((1−2b)a^2q/(q⁻¹⁾A)^1/C andb:=2q/((q−1)C)withC:=1−2bq(q−1)⁻¹. In short, by choosingq large the exponentb >2/(1−2b)becomes arbitrarily close to 2/(1−2b), at the expense of course of a large constantc(b, A, q), sinceAwill have to be chosen sufficiently large.

We sum up these steps into the following simplified version of Theorem1.7.

Corollary 1.8. Ifα=1/2andL(·)is of logarithmic type withb∈(−∞,1/2)(cf.Remark1.1)thenh_c(β) > h_c(0) for everyβ >0and for everyb >2/(1−2b)there existsc >0such that,forβsufficiently small

h_c(β)−h_c(0)≥exp

−cβ⁻^b

. (1.27)

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This result of course has to be compared with the upper bound in Theorem1.5that for L(·)of logarithmic type yields forb<1/2

h_c(β)−h_c(0)≤C₁β⁻^2b/(1⁻^2b)exp

−C₂β⁻^2/(1⁻^2b)

, (1.28)

whereC₁ andC₂ are positive constants that depend (explicitly) on a,b and on the two constants C₁ andC₂ of Theorem1.5(we stress thatC1>0 andC2>0 for everya>0 andb<1/2).

The main body of the proof of Theorem1.7is given in the next section. In the subsequent sections a number of technical results are proven. In the last section (Section6) we prove a general result (Proposition6.1) for the models we are considering: the monotonicity of the free energy with respect toβ. This result, already known for other disordered models, appears not to have been pointed out up to now for the pinning model. We stress that Proposition6.1is not used in the rest of the paper, but, as discussed in Section6, one can find a link of some interest with our main results.

2. Coarse graining, fractional moment and measure change arguments

The purpose of this section is to reduce the proof to a number of technical statements, that are going to be proven in the next sections. In doing so, we are going to introduce the quantities and notations used in the technical statements and, at the same time, we will stress the main ideas and the novelties with respect to earlier approaches (notably, with respect to [14]).

We anticipate that the main ingredients of the proof are (like in [14]) a coarse graining procedure and a fractional moment estimate on the partition function combined with a change of measure. However:

(1) In [14] we have exploited the Gaussian character of the disorder to introduceweak, long-range correlationswhile keeping the Gaussian character of the random variables. In fact, the change of measure is given by a density that is just the exponential of a quadratic functional ofω, that is a measure change via a2-body potential. In order to lower the exponent 4 in the right-hand side of (1.19) we will useq-body potentialsq=3,4, . . .(this is theq appearing in Theorem1.7). Such potentials carry with themselves a number of difficulties: for example, when the law of the disorder is Gaussian, the modified measure is not. As a matter of fact, there are even problems in defining the modified disorder variables if one modifies in a straightforward way the procedure in [14] to useq-body potentials, due to integrability issues: such problems may look absent if one deals with boundedω variables, but they actually reappear when taking limits. The change-of-measure procedure is therefore performed by introducingq-body potentialsandsuitable cut-offs. Estimating the effect of suchq-body potential with cut-off change of measure is at the heart of our technical estimates.

(2) The coarse-graining procedure is different from the one used in [14,20], since we have to adapt it to the new change of measure procedure. However, unlike point (1), the difference between the previous coarse graining procedure and the one we are employing now is more technical than conceptual.

2.1. The coarse graining length

Recall the definition (1.14) of L(·). We are assuming (1.20), therefore limx→∞L(x)= +∞. Chosen a value of q∈ {2,3, . . .}(q is kept fixed throughout the proof) and a positive constantA(that is going to be chosen large) we define

k=k(β;q, A):=inf

n∈N: L(n)/L(n)^2/(q⁻¹⁾≥Aβ⁻^2q/(q⁻¹⁾

. (2.1)

Since we are interested also in cases in whichL(·)diverges (and possibly faster thanL(·)) it is in general false that k <∞. However, the assumption (1.20) guarantees that, forq > (2ε)⁻¹,L(x)/L(x)^2/(q⁻¹⁾→ ∞for x→ ∞ and thereforek <∞.

Moreover, if L(·) is of logarithmic type (Remark 1.1) with b<1/2, then for q >1/(1−2b) the function L(·)/L(·)^2/(q⁻¹⁾is (eventually) increasing.

Of coursek(β;q, A)is just 1/(β;q, A), cf. (1.23), and the reason for such a link is explained in Remark2.5.

Note by now thatkis monotonic in bothβ andA. Sinceβ is chosen smaller than an arbitrary fixed quantityβ0, in order to guarantee thatkis large we will rather play on choosingAlarge.

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Remark 2.1. For the proof certain monotonicity properties will be important.Notably,we know[4],Section1.5.2, that1/(√

xL(x))is asymptotic to a monotonic(decreasing)function and this directly implies that we can find a slowly varying functionL(·)and a constantc_L∈(0,1]such that

x→ 1

√xL(x)is decreasing and cLL(x)≤L(x)≤L(x) for everyx∈(0,∞). (2.2) Given the asymptotic behavior of the renewal function ofτ (a special case of(1.12))

P(n∈τ )ⁿ^→∞∼ 1 2π√

nL(n), (2.3)

and the fact thatP(n∈τ ) >0for everyn∈N,we can chooseL(·)andc_Lsuch that we have also

√ 1

n+1L(n+1)≤P(n∈τ )≤ c⁻_L¹

√n+1L(n+1), n=0,1,2, . . . . (2.4)

It is natural to chooseL(·)such thatlimx→∞L(x)/L(x)∈ [1,1/cL)exists,and we will do so.For later convenience we set

R_1/2(x):= 1

√x+1L(x+1). (2.5)

2.2. The coarse graining procedure and the fractional moment bound Let us start by introducing for 0≤M < Nthe notation

ZM,N=ZM,N,ω:=E

e^N^n=M+1^(βωⁿ⁺^h⁻^logM(β))δnδN|δM=1

, (2.6)

andZ_M,M:=1 (of courseZ_N,ω=Z_0,N). We consider without loss of generality a system of size proportional tok, that isN=kmwithm∈N. ForI⊂ {1, . . . , m}we define

Z^I_ω:=E

e^Nⁿ⁼¹^(βωⁿ⁺^h⁻^logM^(β))δnδ_N1E_I(τ )

, (2.7)

whereE_I:= {τ∩(

i∈IBi)=τ\ {0}}, and Bi:=

(i−1)k+1, . . . , ik

, (2.8)

that isE_Iis the event that the renewalτ intersects the blocks(B_i)_i_∈Iand only these blocks over{1, . . . , N}. It follows from this definition that

Z_N,ω=

I⊂{1,...,m}

Z_ω^I. (2.9)

Note thatZ^I_ω=0 ifm /∈I. Therefore in the following we will always assumem∈I. ForI= {i₁, . . . , i_l}, (i1<· · ·<

il,il=m), one can expressZ_ω^Iin the following way:

Z^I_ω=

d₁,f₁∈B_i₁ d1≤f1

d₂,f₂∈B_i₂ d2≤f2

· · ·

d_l∈B_il

K(d₁)z_d₁Z_d₁_,f₁K(d₂−f₁)z_d₂Z_d₂_,f₂· · ·K(d_l−f_l₋₁)z_d_lZ_d_l_,N, (2.10)

withz_n:=exp(βωn+h−log M(β)). Let us fix a value ofγ∈(0,1)(we actually chooseγ=6/7, but we will keep writing it asγ). Using the inequality(

a_i)^γ ≤

a_i^γ(which is valid fora_i≥0 and an arbitrary collection of indexes) we get

E Z_N,ω^γ

≤

I⊂{1,...,m}

EZ_ω^Iγ

. (2.11)

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An elementary, but crucial, observation is that

F(β, h)= lim

N→∞

1

γ NElogZ_N,ω^γ ≤lim inf

N→∞

1

γ NlogEZ^γ_N,ω, (2.12)

so that if we can prove that lim sup_NEZ^γ_N,ω<∞forh=hc(0)+(β;q, A)we are done.

2.3. The change of measure We introduce

X_j:=

i∈B_j^q

V_k(i)ω_i, (2.13)

whereB_j^q is the Cartesian product ofB_j with itselfq times andω_i=q

a=1ω_i_a. ThepotentialV_k(·)plays a crucial role for the sequel: we define it and discuss some of its properties in the next remark.

Remark 2.2. The potential V is best introduced if we define the sorting operator s(·): if i∈R^q (q =2,3, . . .), s(i)∈R^qis the non-decreasing rearrangement of the entries ofi.We introduce then

U (i):=

q

a=2

R_1/2

s(i)_a−s(i)_a₋₁

, (2.14)

The potentialV is defined by renormalizingUand by setting to zero the diagonal terms:

Vk(i):= 1

(q!)^1/2k^1/2L(k)^(q⁻^1)/2U (i)1_{i_a=i_bfor everya,b}, (2.15) whereL(·)is defined as in (1.14),withL(·)replaced by L(·).By exploiting the fact that for everyc >0 we have

i≤cNR1/2(i)²^N^→∞∼ L(N )one sees that

i∈B₁^q

V_k(i)²= 1 k(L(k))^q⁻¹

0<i1<···<iq≤k

q

a=2

R1/2(i_a−i_a₋1)2k→∞∼ 1. (2.16)

Therefore,

i∈B₁^q

Vk(i)²≤2 (2.17)

forksufficiently large.

Let us introduce, forK >0, also f_K(x):= −K1_{_x_≥_exp(K2)}, g_I(ω):=exp

j∈I

f_K(X_j)

, (2.18)

¯

g(ω):=exp

f_K(X₁) .

We are now going to replace, for fixedI, the measureP(dω)withg_I(ω)P(dω). The latter is not a probability measure:

we could normalize it, but this is inessential because we are directly exploiting Hölder inequality to get EZ_ω^Iγ

≤ E

g_I(ω)⁻^{γ /(1}⁻^{γ )}1−γ E

g_I(ω)Z^I_ωγ

. (2.19)

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The first factor in the right-hand side is easily controlled, in fact E

g_I(ω)⁻^{γ /(1}⁻^{γ )}

=E

¯

g(ω)⁻^{γ /(1}⁻^{γ )}_|I|

=

exp Kγ

1−γ

−1

P

X1≥exp K²

+1 _|I|

, (2.20)

and sinceX1is centered and its variance coincides with the left-hand side of (2.17), by Chebyshev inequality the term exp(Kγ /(1−γ ))P(X₁≥exp(K²))can be made arbitrarily small by choosingKlarge. Therefore forKsufficiently large (depending only onγ (=6/7))

EZ_ω^Iγ

≤2^γ^|I|

E

g_I(ω)Z_ω^Iγ

. (2.21)

Estimating the remaining factor is a more involved matter. We will actually use the following two statements, that we prove in the next section. SetP_I:=P(E_I;δN=1).

Proposition 2.3. Assume thatα=1/2and that(1.20)holds for someε∈(0,1/2].For everyη >0and everyq >

(2ε)⁻¹we can chooseA >0such that ifβ≤β0andh≤(β;q, A),for everyI⊂ {1, . . . , m}withm∈Iwe have E

g_I(ω)Z_ω^I

≤η^|I|P_I. (2.22)

The following technical estimate controlsP_I (recall thatI= {i₁, . . . , i_|I|}).

Lemma 2.4. Assumeα=1/2.There existC₁=C₁(L(·), k),C₂=C₂(L(·))andk₀=k₀(L(·))such that(withi₀:=0)

P_I≤C1C₂^|I|

|I|

j=1

1

(i_j−i_j₋₁)^7/5 (2.23)

fork≥k₀.

Note that in this statementkis just a natural number, but we will apply it withkas in (2.1) so thatk≥k0is just a requirement onA. Note also that the choice of 7/5 is arbitrary (any number in(1,3/2)would do: the constantsC₁ andC₂depend on such a number).

Let us now go back to (2.21) and let us plug it into (2.11) and use Proposition2.3and Lemma2.4to get:

E Z_N,ω^γ

≤C₁^γ

I⊂{1,...,m} m∈I

|I|

j=1

(2C2η)^γ (i_j−i_j₋₁)^{7γ /5}

. (2.24)

But 7γ /5=6/5>1, so we can choose

η:= 1

3C2(_∞

i=1i⁻^6/5)^7/6, (2.25)

and this implies thatn→(2C2η)^γn⁻^{7γ /5}is a sub-probability, which directly entails that E

Z_N,ω^γ

≤C₁^γ (γ =6/7) (2.26)

for everyN, which implies, via (2.12), thatF(β, h)=0 and we are done.

It is important to stress thatC₁may depend onk(we need (2.26) uniform inN, not ink), butC₂does not (C2is just a function ofL(·), that is a function of the chosen renewal), so thatηmay actually be chosen a priori as in (2.25):

it is a small but fixed constant that depends only on the underlying renewalτ.

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Fig. 2. The figure above explains our coarse graining procedure. HereN=8k,I= {2,5,6,8}. The drawn trajectory is a typical trajectory con- tributing toZ_N,ω^I ;d_i andf_i, 1≤i≤4, correspond to the indexes of (2.10). The shadowed regions represent the sites on which the change of measure procedure (presented in Section2.3) acts.

Remark 2.5. In this section we have actually hidden the role of (β;q, A) in the hypotheses of Proposition2.3, which are the hypotheses of Theorem1.7.Let us therefore explain informally why we can prove a critical point shift of=1/k.

The coarse graining procedure reduces proving delocalization to Proposition2.3.As it is quite intuitive from(2.9)–

(2.21)and Fig.2,one has to estimate the expectation,with respect to the g(ω)-modified measure,¯ of the partition functionZ_d_j_,f_j (or,equivalently,Z_d_j_,f_j/P(fj−d_j∈τ ))in each visited block(let us assume thatf_j −d_j is of the order ofk,because if it is much smaller thankone can bound this contribution in a much more elementary way).The Boltzmann factor inZd_j,f_j isexp(fj

n=dj+1(βωn−log M(β)+h)δn)which can be bounded(in an apparently very rough way)byexp(fj

n=dj+1(βω_n−log M(β))δn)exp(hk),sincef_j−d_j≤k.Therefore,ifh≤(β;q, A)∼1/k we can drop the dependence onhat the expense of the multiplicative factorethat is innocuous because we can show that the expectation(with respect to theg(ω)-modified measure)¯ ofZ_d_j_,f_j/P(fj−d_j∈τ )whenh=0can be made arbitrarily small by choosingAsufficiently large.

Remark 2.6. How does one guess the change of measure in(2.18)? An argument that suggests how to choose the den- sityg_I(ω)goes as follows.The reason whyE[Z_N,ω]does not capture the correct typical behavior ofZ_N,ωforNlarge lies in the atypically large values ofZN,ωthat dominate the expectation.In order to get closer to the typical behavior one can introduce a penalization in terms of a density likeexp(fK(Z_N,ω)).However this change of measure is very difficult to handle,because it essentially requires being able to compute quantities involvingZ_N,ω,that is precisely the object that we are investigating.So what we do is to attempt the substitution ofZ_N,ωwith a much simpler quantity that is expected to capture some crucial features of it.A naive,but in the end fruitful,way of proceeding is to expand the Boltzmann factorexp(H )=1+H+H²/2+ · · ·.If one considers the term of orderq one has to take theqth power of_N

n=1(βω_n+h−log M(β))δnthus obtaining in particularβ^q

i1<i2<···<iqω_i₁ω_i₂· · ·ω_i_qδ_i₁δ_i₂· · ·δ_i_q.By replacingβ^qδ_i₁δ_i₂· · ·δ_i_q by its expectation one gets essentiallyV_k(i₁, . . . , i_q)(cf. (2.15))forN=k:this can be seen by using the asymptotic behavior ofE[δ_i],that is(2.3),and the definition ofkthat tells us thatL(k)/L(k)^(q⁻^1)/2is aboutβ^q(in this rough approximation procedure we have replacedi1withk:this is because in most of the terms of the sumi₁is actually of this order of magnitude).

Remark 2.7. A last observation on the proof is aboutβ₀.It can be chosen arbitrarily,but for the sake of simplifying the constants appearing in the proofs we chooseβ₀∈(0,∞)such that

1 2 ≤ d²

dβ²log M(β)≤2 (2.27)

forβ∈ [0, β0].Choosingβ0arbitrarily just boils down to changing the constants in the right-most and left-most terms in(2.27).

3. Coarse graining estimates

We start by proving Lemma2.4, namely (2.23). The proof is however more clear if instead of working with the exponent 7/5 we work with 3/2−ξ (ξ ∈(0,1/2), in the end, plug inξ=1/10).