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Hierarchical pinning model in correlated random environment

Abstract : 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.
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Contributor : Quentin Berger <>
Submitted on : Wednesday, January 4, 2017 - 1:21:52 PM
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Quentin Berger, Fabio Toninelli. Hierarchical pinning model in correlated random environment. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2013, 49, pp.781 - 816. ⟨10.1214/12-AIHP493⟩. ⟨hal-01426311⟩

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