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18
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.
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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Cited by 22 (4 self)
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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.
The martingale approach to disorder irrelevance for pinning models, Electron
 Comm. Probab
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HIERARCHICAL PINNING MODEL WITH SITE DISORDER: DISORDER IS MARGINALLY RELEVANT
, 2008
"... We study a hierarchical disordered pinning model with site disorder for which, like in the bond disordered case [5, 8], there exists a value of a parameter b (enters in the definition of the hierarchical lattice) that separates an irrelevant disorder regime and a relevant disorder regime. We show ..."
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Cited by 9 (6 self)
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We study a hierarchical disordered pinning model with site disorder for which, like in the bond disordered case [5, 8], there exists a value of a parameter b (enters in the definition of the hierarchical lattice) that separates an irrelevant disorder regime and a relevant disorder regime. We show that for such a value of b the critical point of the disordered system is different from the critical point of the annealed version of the model. The proof goes beyond the technique used in [8] and it takes explicitly advantage of the inhomogeneous character of the Green function of the model.
M. Philippe CARMONA Universite ́ de Nantes rapporteur M. Francis COMETS Universite ́ ParisDiderot rapporteur
"... Bernard Derrida, Giambattista Giacomin et Hubert Lacoin. Je travaille avec eux avec le plus grand plaisir, et j’apprécie profondément leurs qualités a ̀ la fois humaines et scientifiques. Les discussions que nous avons eu ensemble m’ont toujours fait découvrir des nouveaux points de vue sur la m ..."
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Bernard Derrida, Giambattista Giacomin et Hubert Lacoin. Je travaille avec eux avec le plus grand plaisir, et j’apprécie profondément leurs qualités a ̀ la fois humaines et scientifiques. Les discussions que nous avons eu ensemble m’ont toujours fait découvrir des nouveaux points de vue sur la mécanique statistique et sur les probabilités. Je tiens aussi a ̀ remercier d’autres collègues avec lesquels j’ai eu des échanges scientifique très importants sur les sujets de ce travail: Ken Alexander, Thierry Bodineau, Erwin Bolthausen, Francesco Caravenna, Frank den Hollander, Krzysztof Gawedzki et
3 THE CRITICAL CURVES OF THE RANDOM PINNING AND COPOLYMER MODELS AT WEAK COUPLING
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SUBCRITICAL PERCOLATION WITH A LINE OF DEFECTS
"... Abstract. We consider the Bernoulli bond percolation process Pp,p ′ on the nearestneighbor edges of Zd, which are open independently with probability p < pc, except for those lying on the first coordinate axis, for which this probability is p′. Define ξp,p ′: = − lim n→∞n −1 logPp,p′(0 ↔ ne1), ..."
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Abstract. We consider the Bernoulli bond percolation process Pp,p ′ on the nearestneighbor edges of Zd, which are open independently with probability p < pc, except for those lying on the first coordinate axis, for which this probability is p′. Define ξp,p ′: = − lim n→∞n −1 logPp,p′(0 ↔ ne1), and ξp: = ξp,p. We show that there exists p c = p c(p, d) such that ξp,p ′ = ξp if p ′ < p′c and ξp,p ′ < ξp if p ′> p′c. Moreover, p′c(p, 2) = p c(p, 3) = p, and p
Hierarchical pinning model: low disorder relevance in the b = s case.
, 2014
"... We consider a hierarchical pinning model introduced by B.Derrida, V.Hakim and J.Vannimenus which undergoes a localization/delocalization phase transition. This model depends on two parameters b and s. We show that in the particular case where b = s, the disorder is weakly relevant, in the sense that ..."
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We consider a hierarchical pinning model introduced by B.Derrida, V.Hakim and J.Vannimenus which undergoes a localization/delocalization phase transition. This model depends on two parameters b and s. We show that in the particular case where b = s, the disorder is weakly relevant, in the sense that at any given temperature, the quenched and the annealed critical points coincide. This is in contrast with the case where b 6 = s.