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49
A replicacoupling approach to disordered pinning models
 Comm. Math. Phys
"... Abstract. We consider a renewal process τ = {τ0, τ1,...} on the integers, where the law of τi − τi−1 has a powerlike tail P(τi − τi−1 = n) ≃ n −(α+1) with α ≥ 0. We then assign a random, ndependent reward/penalty to the occurrence of the event that the site n belongs to τ. In such generality this ..."
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Cited by 37 (4 self)
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Abstract. We consider a renewal process τ = {τ0, τ1,...} on the integers, where the law of τi − τi−1 has a powerlike tail P(τi − τi−1 = n) ≃ n −(α+1) with α ≥ 0. We then assign a random, ndependent reward/penalty to the occurrence of the event that the site n belongs to τ. In such generality this class of problems includes, among others, (1 + d)dimensional models of pinning of directed polymers on a onedimensional random defect, (1 + 1)dimensional models of wetting of disordered substrates, and the PolandScheraga model of DNA denaturation. By varying the average of the reward, the system undergoes a transition from a localized phase where τ occupies a finite fraction of N to a delocalized phase where the density of τ vanishes. In absence of disorder (i.e., if the reward is nindependent), the transition is of first order for α> 1 and of higher order for α < 1. Moreover, for α ranging from 1 to 0, the transition ranges from first to infinite order. Presence of even an arbitrarily small amount of disorder is known to modify the order of transition as soon as α> 1/2 [11]. In physical terms, disorder is relevant in this situation, in agreement with the heuristic Harris criterion. On the other hand, for α < 1/2 it has been proven recently by K. Alexander [2] that, if disorder is sufficiently weak, critical exponents are not modified by randomness: disorder is irrelevant. In this work, applying techniques which in the framework of spin glasses are known as replica coupling and interpolation, we give a new, simpler proof of the main results of [2].
Fractional moment bounds and disorder relevance for pinning models
 COMMUN. MATH. PHYS
, 2007
"... We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n −α−1 L(n), with L(·) slowly varying. The model undergoes a (de)locali ..."
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Cited by 36 (16 self)
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We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n −α−1 L(n), with L(·) slowly varying. The model undergoes a (de)localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For α < 1/2 it is known that disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [2, 24]. The same has been proven also for α = 1/2, but under the assumption that L(·) diverges sufficiently fast at infinity, an hypothesis that is not satisfied in the (1 + 1)dimensional wetting model considered in [13, 9], where L(·) is asymptotically constant. Here we prove that, if 1/2 < α < 1 or α> 1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the socalled Harris criterion, disorder is therefore relevant in this case. In the marginal case α = 1/2, under the assumption that L(·) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed
Quenched and annealed critical points in polymer pinning models
"... Abstract. We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential u+Vn which the chain encounters when it visits a special state 0 at time n. The disorder (Vn) is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spe ..."
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Cited by 34 (6 self)
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Abstract. We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential u+Vn which the chain encounters when it visits a special state 0 at time n. The disorder (Vn) is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when u exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length n has the form n −c ϕ(n) with c ≥ 1 and ϕ slowly varying. Comparing to the corresponding annealed system, in which the Vn are effectively replaced by a constant, it was shown in [1], [4], [11] that the quenched and annealed critical points differ at all temperatures for 3/2 < c < 2 and c> 2, but only at low temperatures for c < 3/2. For high temperatures and 3/2 < c < 2 we establish the exact order of the gap between critical points, as a function of temperature. For the borderline case c = 3/2 we show that the gap is positive provided ϕ(n) → 0 as n → ∞, and for c> 3/2 with arbitrary temperature we provide an alternate proof of the result in [4] that the gap is positive, and extend it to c = 2. 1.
DISORDERED PINNING MODELS AND COPOLYMERS: BEYOND Annealed Bounds
, 2008
"... We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by T. Garel et al. [16], pinning and wetting models in various dimensions, and the PolandScheraga model of DNA denaturati ..."
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Cited by 32 (4 self)
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We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by T. Garel et al. [16], pinning and wetting models in various dimensions, and the PolandScheraga model of DNA denaturation. We prove a new variational upper bound for the free energy via an estimation of noninteger moments of the partition function. As an application, we show that for strong disorder the quenched critical point differs from the annealed one, e.g., if the disorder distribution is Gaussian. In particular, for pinning models with loop exponent 0 < Î± < 1/2 this implies the existence of a transition from weak to strong disorder. For the copolymer model, under a (restrictive) condition on the law of the underlying renewal, we show that the critical point coincides with the one predicted via renormalization group arguments in the theoretical physics literature. A stronger result holds for a âreduced wetting model â introduced by T. Bodineau and G. Giacomin [5]: without restrictions on the law of the underlying renewal, the critical point coincides with the corresponding renormalization group prediction.
Marginal relevance of disorder for pinning models
, 2009
"... The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics (e.g. [16, 11]). 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 exp ..."
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Cited by 29 (10 self)
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The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics (e.g. [16, 11]). 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 α, 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 α < 1/2. Recent mathematical work (in particular [2, 10, 21, 22]) has put these predictions on firm grounds. In renormalization group terms, the case α = 1/2 is a marginal case and there is no agreement in the literature as to whether one should expect disorder relevance [11] or irrelevance [16] at marginality. The question is particularly intriguing also because the case α = 1/2 includes the classical models of twodimensional 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 threedimensional space. Here we prove disorder relevance both for the general α = 1/2 pinning model and for the hierarchical version of the model proposed in [11], 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/β 4) for β small, if β 2 is the disorder variance.
Pinning and wetting transition for (1+1)dimensional fields with Laplacian interaction
 Ann. Probab
"... Abstract. We consider a random field ϕ: {1,..., N} → R as a model for a linear chain attracted to the defect line ϕ = 0, i.e. the x–axis. The free law of the field is specified by the density exp ` − P i V (∆ϕi) ´ with respect to the Lebesgue measure on R N, where ∆ is the discrete Laplacian and w ..."
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Cited by 25 (4 self)
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Abstract. We consider a random field ϕ: {1,..., N} → R as a model for a linear chain attracted to the defect line ϕ = 0, i.e. the x–axis. The free law of the field is specified by the density exp ` − P i V (∆ϕi) ´ with respect to the Lebesgue measure on R N, where ∆ is the discrete Laplacian and we allow for a very large class of potentials V (·). The interaction with the defect line is introduced by giving the field a reward ε ≥ 0 each time it touches the x–axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity ε of the pinning reward varies: both in the pinning (a = p) and in the wetting (a = w) case, there exists a critical value ε a c such that when ε> ε a c the field touches the defect line a positive fraction of times (localization), while this does not happen for ε < ε a c (delocalization). The two critical values are nontrivial and distinct: 0 < ε p c < ε w c < ∞, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at ε = ε p c is delocalized. On the other hand, the transition in the wetting model is of first order and for ε = ε w c the field is localized. The core of our approach is a Markov renewal theory description of the field. 1. Introduction and
Disorder relevance at marginality and critical point shift
 Ann. Inst. Henri Poincar e Probab. Stat
"... 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 relevant or irrelevant in the Harris criterion sense: the question addressed is whether quenched disord ..."
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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 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 [14] we have proven that the two critical points differ at marginality of at least exp(−c/β4), where c> 0 and β2 is the disorder variance, for β ∈ (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)/βb), c(b)> 0 for b> 2 (b = 2 is the known upper bound and it is the result claimed in [8]), 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 multibody potential modifications of the law of the disorder.
EQUALITY OF CRITICAL POINTS FOR POLYMER DEPINNING TRANSITIONS WITH LOOP EXPONENT ONE
, 811
"... Abstract. We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u + Vn when it visits a particular state 0 at time n, with {Vn} representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned ..."
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Cited by 18 (2 self)
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Abstract. We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u + Vn when it visits a particular state 0 at time n, with {Vn} representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned by the potential. A particular case not covered in a number of previous studies is that of loop exponent one, in which the probability of an excursion of length n takes the form ϕ(n)/n for some slowly varying ϕ; this includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of u in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ at least at low temperatures. 1.
The martingale approach to disorder irrelevance for pinning models, Electron
 Comm. Probab
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On the approach to equilibrium for a polymer with adsorption and repulsion, Electronic Journal of Probability 13
, 2008
"... Abstract. We consider paths of a one–dimensional simple random walk conditioned to come back to the origin after L steps, L ∈ 2N. In the pinning model each path η has a weight λ N(η) , where λ> 0 and N(η) is the number of zeros in η. When the paths are constrained to be non–negative, the polymer ..."
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Cited by 14 (5 self)
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Abstract. We consider paths of a one–dimensional simple random walk conditioned to come back to the origin after L steps, L ∈ 2N. In the pinning model each path η has a weight λ N(η) , where λ> 0 and N(η) is the number of zeros in η. When the paths are constrained to be non–negative, the polymer is said to satisfy a hard– wall constraint. Such models are well known to undergo a localization/delocalization transition as the pinning strength λ is varied. In this paper we study a natural “spin flip ” dynamics for these models and derive several estimates on its spectral gap and mixing time. In particular, for the system with the wall we prove that relaxation to equilibrium is always at least as fast as in the free case (i.e. λ = 1 without the wall), where the gap and the mixing time are known to scale as L −2 and L 2 log L, respectively. This improves considerably over previously known results. For the system without the wall we show that the equilibrium phase transition has a clear dynamical manifestation: for λ � 1 relaxation is again at least as fast as the diffusive free case, but in the strictly delocalized phase (λ < 1) the gap is shown to be O(L −5/2), up to logarithmic corrections. As an application of our bounds, we prove stretched exponential relaxation of local functions in the localized regime.