Hierarchical pinning model in correlated random environment

TL;DR

This paper investigates how spatially correlated disorder affects the phase transition in a hierarchical pinning model, revealing three distinct regimes based on correlation decay, including the disappearance or modification of the transition.

Contribution

It extends the understanding of disordered pinning models by analyzing the impact of spatial correlations on critical behavior in a hierarchical setting.

Findings

Disappearance of phase transition with non-summable correlations.

System behaves like i.i.d. case with fast-decaying correlations.

Correlations can alter critical properties of the annealed system.

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.