Disorder relevance without Harris Criterion: the case of pinning model with $\gamma$-stable environment

TL;DR

This paper explores how disorder affects the pinning of a renewal process in a stable environment, revealing a new criterion for disorder relevance that differs from the Harris criterion.

Contribution

It introduces a new criterion for disorder relevance in pinning models with stable environments, beyond the traditional Harris criterion.

Findings

Disorder causes a shift in the critical point when >1-mma^{-1}.

At high temperature, quenched and annealed critical points coincide for < 1-mma^{-1}.

Critical exponents are identical in the latter case.

Abstract

We investigate disorder relevance for the pinning of a renewal whose inter-arrival law has tail exponent $\alpha>0$ when the law of the random environment is in the domain of attraction of a stable law with parameter $\gamma \in (1,2)$. We prove that in this case, the effect of disorder is not decided by the sign of the specific heat exponent as predicted by Harris criterion but that a new criterion emerges to decide disorder relevance. More precisely we show that when $\alpha>1-\gamma^{-1}$ there is a shift of the critical point at every temperature whereas when $\alpha< 1-\gamma^{-1}$, at high temperature the quenched and annealed critical point coincide, and the critical exponents are identical.