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The presence of frozen-in or quenched disorder in a system can often modify the nature of its phase transition. A particular instance of this phenomenon is the so-called rounding effect: it has been shown in many cases that the free-energy curve of the disordered system at its critical point is smoother than that of the homogenous one. In particular some disordered systems do not allow first-order transitions. We study this phenomenon for the pinning of a renewal with stretched-exponential tails on a defect line (the distribution $K$ of the renewal increments satisfies $K(n) \sim c_K\exp(-n^{\alpha}),$ $\alpha\in (0,1)$) which has a first order transition when disorder is not present. We show that the critical behavior of the disordered system depends on the value of $\alpha$: when $\alpha>1/2$ the transition remains first order, whereas the free-energy diagram is smoothed for $\alpha\le 1/2$. Furthermore we show that the rounding effect is getting stronger when $\alpha$ diminishes., Comment: 20 pages, 2 Figure, a few minor errors corrected

We investigate disorder relevance for the pinning of a renewal whose inter-arrival law has tail exponent $\alpha >0$ when the law of the random environment is in the domain of attraction of a stable law with parameter $\gamma \in (1,2)$. We prove that in this case, the effect of disorder is not decided by the sign of the specific heat exponent as predicted by Harris criterion but that a new criterion emerges to decide disorder relevance. More precisely we show that when $\alpha >1-\gamma ^{-1}$ there is a shift of the critical point at every temperature whereas when $\alpha < 1-\gamma ^{-1}$, at high temperature the quenched and annealed critical points coincide, and the critical exponents are identical.

The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point for a small amount of disorder. The question has been mathematically settled in most cases in the last few years, giving in particular a rigorous validation of the Harris Criterion on disorder relevance. However, the marginal case, where the return probability exponent is equal to $1/2$, i.e. where the inter-arrival law of the renewal process is given by $K(n)=n^{-3/2}\phi(n)$ where $\phi$ is a slowly varying function, has been left partially open. In this paper, we give a complete answer to the question by proving a simple necessary and sufficient criterion on the return probability for disorder relevance, which confirms earlier predictions from the literature. Moreover, we also provide sharp asymptotics on the critical point shift: in the case of the pinning (or wetting) of a one dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptotics $$ \lim_{\beta\rightarrow 0}\beta^2\log h_c(\beta)= - \frac{\pi}{2}. $$ This gives a rigorous proof to a claim of B. Derrida, V. Hakim and J. Vannimenus (Journal of Statistical Physics, 1992)., Comment: 34 Pages

We consider the hierarchical disordered pinning model studied in [9], which exhibits a localization/delocalization phase transition. In the case where the disorder is i.i.d. (independent and identically distributed), the question of relevance/irrelevance of disorder (i.e. whether disorder changes or not the critical properties with respect to the homogeneous case) is by now mathematically rather well understood [14,15]. Here we consider the case where randomness is spatially correlated and correlations respect the hierarchical structure of the model; in the non-hierarchical model our choice would correspond to a power-law decay of correlations. In terms of the critical exponent of the homogeneous model and of the correlation decay exponent, we identify three regions. In the first one (non-summable correlations) the phase transition disappears. In the second one (correlations decaying fast enough) the system behaves essentially like in the i.i.d. setting and the relevance/irrelevance criterion is not modified. Finally, there is a region where the presence of correlations changes the critical properties of the annealed system., Comment: 38 pages, 3 figures

We study the random pinning model, in the case of a Gaussian environment presenting power-law decaying correlations, of exponent decay a>0. We comment on the annealed (i.e. averaged over disorder) model, which is far from being trivial, and we discuss the influence of disorder on the critical properties of the system. We show that the annealed critical exponent ��^{ann} is the same as the homogeneous one ��^{pur}, provided that correlations are decaying fast enough (a>2). If correlations are summable (a>1), we also show that the disordered phase transition is at least of order 2, showing disorder relevance if ��^{pur}, 23 pages, 1 figure Modifications in v2 (outside minor typos): Assumption 1 on correlations has been simplified for more clarity; Theorem 4 has been improved to a more general underlying renewal distribution; Remark 2.1 added, on the assumption on the correlations in the summable case

In this dissertation we study the pinning model with weakly correlated disorder.The pinning model applies to various situations such as localization of a polymernear a one-dimensional interface, wetting transition and DNA denaturation, whichall display a transition between a localized phase and a delocalized phase.We start by giving a survey of the available results concerning critical pointsand exponents, first for the homogeneous setup and then for the inhomogeneousone, in the case when disorder is given by a sequence of independent and identicallydistributed (i.i.d.) random variables. In the latter case, we also provide a hightemperaturebound on the quenched critical curve in a case of relevant disorder.We then study the random pinning model when disorder is gaussian and hascorrelations with finite range, using the theory of Markov renewal processes. Weexpress the annealed critical curve in terms of the largest eigenvalue of a transfermatrix and we give the annealed critical exponent. We then generalize the criteriafor disorder relevance/irrelevance that were proved for the i.i.d. case.Next we are interested in disorder sequences with infinite range correlations.At first we generalize the method used to deal with finite range correlations andobtain the annealed critical behaviour in the case of gaussian disorder assumingfast decay of correlations. We use to this end the spectral properties of transferoperators for shifts on integer sequences and potentials with summable variations.Secondly we provide some results when disorder is a Markov chain.; Cette thèse est consacrée à l’étude du modèle d’accrochage en environnementfaiblement corrélé. Le modèle d’accrochage s’applique à de multiples situationstelles que la localisation d’un polymère au voisinage d’une interface unidimensionnelle,la transition de mouillage ou encore la dénaturation de l’ADN, le pointcommun étant la présence d’une transition entre une phase localisée et une phasedélocalisée.Nous commençons par donner un aperçu des résultats disponibles sur lescourbes et exposants critiques pour le modèle homogène puis pour le modèledésordonné lorsque le désordre est une suite de variables aléatoires indépendanteset identiquement distribuées (i.i.d.). Dans ce dernier cas, nous donnons égalementune borne sur la courbe critique quenched à haute température, dans un régimeoù le désordre est dit pertinent.Nous étudions ensuite le modèle d’accrochage désordonné dans le cas où ledésordre est gaussien et les corrélations ont une portée finie, à l’aide de la théoriedes processus de renouvellement markoviens. Nous donnons dans ce cas une expressionde la courbe annealed à l’aide de la plus grande valeur propre d’une matricede transfert ainsi que l’exposant critique annealed. Nous généralisons ensuite lescritères de pertinence et de non pertinence du désordre prouvés dans le cas i.i.d.Nous nous intéressons ensuite à des désordres dont les corrélations ont uneportée de corrélation infinie. Dans un premier temps, nous généralisons la démarcheutilisée dans le cas d’une portée de corrélations finie et obtenons le comportementcritique annealed dans le cas d’un désordre gaussien sous des hypothèses dedécroissance forte des corrélations. Nous utilisons pour cela les propriétés spectralesdes opérateurs de transfert pour des décalages sur des suites d’entiers etdes potentiels à variations sommables. Dans un deuxième temps, nous donnonsquelques résultats dans le cas où le désordre est donné par une chaîne de Markov.

The purpose of this paper is to show how one can extend some results on disorder relevance obtained for the random pinning model with i.i.d disorder to the model with finite range correlated disorder. In a previous work, the annealed critical curve of the latter model was computed, and equality of quenched and annealed critical points, as well as exponents, was proved under some conditions on the return exponent of the interarrival times. Here we complete this work by looking at the disorder relevant regime, where annealed and quenched critical points differ. All these results show that the Harris criterion, which was proved to be correct in the i.i.d case, remains valid in our setup. We strongly use Markov renewal constructions that were introduced in the solving of the annealed model.

This thesis studies models of polymers in random environment: we focus on the case of a directed polymer in dimension d+1 that interacts with a one-dimensional defect. The interactions are possibly inhomogeneous, and are represented by random variables. We deal with the question of the influence of disorder on the localization phenomenon: we want to determine if the presence of inhomogeneities modifies the critical properties of the system, and especially the characteristics of the phase transition (in that case disorder is said to be pertinent). In particular, we prove that if the defect is a random walk, disorder is relevant in dimension d≥3. We then study the pinning model in random correlated environment. There is a non-rigourous criterion (due to Weinrib and Halperin), that we can apply to our model, and that predicts disorder relevance/irrelevance, according to the value of the critical exponent of the homogeneous system, denoted νpur, and of the correlation decay exponent. When disorder is Gaussian and correlations are summable, we show that the Weinrib-Halperin criterion is valid: we prove this in the hierarchical version of the model, and also, partially, in the non-hierachical (standard) framework. Moreover, we obtained a surprising result: when correlations are sufficiently strong, and in particular when they are non-summable (in the gaussian framework), a new regime in which disorder is always relevant appears, the order of the phase transition being always larger than νpur. The Weinrib-Halperin prediction therefore does not apply to our model., Cette thèse porte sur l'étude de modèles de polymère en milieu aléatoire: on se concentre sur le cas d'un polymère dirigé en dimension d+1 qui interagit avec un défaut unidimensionnel. Les interactions sont possiblement non-homogènes, et sont représentées par des variables aléatoires. Une question importante est celle de l'influence du désordre sur le phénomène de localisation: on veut déterminer si la présence d'inhomogénéités modifie les propriétés critiques du système, et notamment les caractéristiques de la transition de phase (auquel cas le désodre est dit pertinent). En particulier, nous prouvons que dans le cas où le défaut est une marche aléatoire, le désordre est pertinent en dimension d≥3. Ensuite, nous étudions le modèle d'accrochage sur une ligne de défauts possédant des inhomogénéités corrélées spatialement. Il existe un critère non rigoureux (dû à Weinrib et Halperin), que l'on applique à notre modèle, et qui prédit si le désordre est pertinent ou non en fonction de l'exposant critique du système homogène, noté νpur, et de l'exposant de décroissance des corrélations. Si le désordre est gaussien et les corrélations sommables, nous montrons la validité du critère de Weinrib-Halperin: nous le prouvons dans la version hiérarchique du modèle, et aussi, de manière partielle, dans le cadre (standard) non-hiérarchique. Nous avons de plus obtenu un résultat surprenant: lorsque les corrélations sont suffisamment fortes, et en particulier si elles sont non-sommables (dans le cadre gaussien), il apparaît un régime où le désordre devient toujours pertinent, l'ordre de la transition de phase étant toujours plus grand que νpur. La prédiction de Weinrib-Halperin ne s'applique alors pas à notre modèle.

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 'relevant' or 'irrelevant' in 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 called 'marginal', 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 a previous work (arXiv:0811.0723) we have proven that the two critical points differ at marginality of at least exp(-c/beta^4), where c>0 and beta^2 is the disorder variance, for beta in (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/beta^4) lower bound on the shift can be replaced by exp(-c(b)/beta^b), c(b)>0 for b>2 (b=2 is the known upper bound and it is the result claimed in [Derrida, Hakim, Vannimenus, JSP 1992]), 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., 30 pages

We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in [5,6,7]. Given a fixed realization of a random walk Y$ on Z^d with jump rate rho (that plays the role of the random medium), we modify the law of a random walk X on Z^d with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time L_t(X,Y)=\int_0^t \ind_{X_s=Y_s}\dd s: the weight of the path under the new measure is exp(beta L_t(X,Y)), beta in R. As beta increases, the system exhibits a delocalization/localization transition: there is a critical value beta_c, such that if beta>beta_c the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d larger or equal to three 3, the presence of disorder makes the phase transition at least of second order. This, in dimension larger or equal to 4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy., 18 pages

Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that is now rigorous. In this work, we exploit interpolation and replica coupling methods to obtain sharper results on the irrelevant disorder regime of pinning models. In particular, in this regime, we compute the first order term in the expansion of the free energy close to criticality and this term coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and quenched averaged correlation length exponents coincide, while, in general, they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals., Comment: Published in at http://dx.doi.org/10.1214/09-AOP454 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics. In particular, it has been predicted on the basis of the Harris criterion that disorder is relevant (annealed and quenched model have different critical points and critical exponents) if the return probability exponent alpha, a positive number that characterizes the model, is larger than 1/2. Weak disorder has been predicted to be irrelevant (i.e. coinciding critical points and exponents) if alpha < 1/2. Recent mathematical work has put these predictions on firm grounds. In renormalization group terms, the case alpha = 1/2 is a 'marginal case' and there is no agreement in the literature as to whether one should expect disorder relevance or irrelevance at marginality. The question is particularly intriguing also because the case alpha = 1/2 includes the classical models of two-dimensional wetting of a rough substrate, of pinning of directed polymers on a defect line in dimension (3+1) or (1+1) and of pinning of an heteropolymer by a point potential in three-dimensional space. Here we prove disorder relevance both for the general alpha = 1/2 pinning model and for the hierarchical version of the model proposed by B. Derrida, V. Hakim and J. Vannimenus (JSP, 1992), in the sense that we prove a shift of the quenched critical point with respect to the annealed one. In both cases we work with Gaussian disorder and we show that the shift is at least of order exp(-1/��^4) for beta small, if beta is the standard deviation of the disorder., 27 pages. v2: section 4.1 simplified. To appear on Commun. Pure Appl. Math