# Disorder relevance at marginality and critical point shift

Giambattista Giacomin; Hubert Lacoin; Fabio Lucio Toninelli

Annales de l'I.H.P. Probabilités et statistiques (2011)

- Volume: 47, Issue: 1, page 148-175
- ISSN: 0246-0203

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topGiacomin, Giambattista, Lacoin, Hubert, and Toninelli, Fabio Lucio. "Disorder relevance at marginality and critical point shift." Annales de l'I.H.P. Probabilités et statistiques 47.1 (2011): 148-175. <http://eudml.org/doc/241051>.

@article{Giacomin2011,

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 [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), 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 [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.},

author = {Giacomin, Giambattista, Lacoin, Hubert, Toninelli, Fabio Lucio},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {disordered pinning models; Harris criterion; marginal disorder; many-body interactions},

language = {eng},

number = {1},

pages = {148-175},

publisher = {Gauthier-Villars},

title = {Disorder relevance at marginality and critical point shift},

url = {http://eudml.org/doc/241051},

volume = {47},

year = {2011},

}

TY - JOUR

AU - Giacomin, Giambattista

AU - Lacoin, Hubert

AU - Toninelli, Fabio Lucio

TI - Disorder relevance at marginality and critical point shift

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2011

PB - Gauthier-Villars

VL - 47

IS - 1

SP - 148

EP - 175

AB - 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 [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), 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 [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.

LA - eng

KW - disordered pinning models; Harris criterion; marginal disorder; many-body interactions

UR - http://eudml.org/doc/241051

ER -

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