Hierarchical pinning model in correlated random environment

Quentin Berger; Fabio Lucio Toninelli

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 3, page 781-816
  • ISSN: 0246-0203

Abstract

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We consider the hierarchical disordered pinning model studied in (J. Statist. Phys.66 (1992) 1189–1213), 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 (Probab. Theory Related Fields148 (2010) 159–175, Pure Appl. Math.63 (2010) 233–265). 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.

How to cite

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Berger, Quentin, and Toninelli, Fabio Lucio. "Hierarchical pinning model in correlated random environment." Annales de l'I.H.P. Probabilités et statistiques 49.3 (2013): 781-816. <http://eudml.org/doc/272100>.

@article{Berger2013,
abstract = {We consider the hierarchical disordered pinning model studied in (J. Statist. Phys.66 (1992) 1189–1213), 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 (Probab. Theory Related Fields148 (2010) 159–175, Pure Appl. Math.63 (2010) 233–265). 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.},
author = {Berger, Quentin, Toninelli, Fabio Lucio},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {pinning models; polymer; disordered models; Harris criterion; critical phenomena; correlation; hierarchical models},
language = {eng},
number = {3},
pages = {781-816},
publisher = {Gauthier-Villars},
title = {Hierarchical pinning model in correlated random environment},
url = {http://eudml.org/doc/272100},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Berger, Quentin
AU - Toninelli, Fabio Lucio
TI - Hierarchical pinning model in correlated random environment
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 3
SP - 781
EP - 816
AB - We consider the hierarchical disordered pinning model studied in (J. Statist. Phys.66 (1992) 1189–1213), 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 (Probab. Theory Related Fields148 (2010) 159–175, Pure Appl. Math.63 (2010) 233–265). 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.
LA - eng
KW - pinning models; polymer; disordered models; Harris criterion; critical phenomena; correlation; hierarchical models
UR - http://eudml.org/doc/272100
ER -

References

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