The random pinning model with correlated disorder given by a renewal set

TL;DR

This paper studies how correlated disorder, modeled by a renewal set, affects the localization transition in a renewal process, extending the Harris criterion and analyzing different correlation regimes.

Contribution

It extends the Harris criterion to correlated disorder given by renewal sets and analyzes disorder relevance across various correlation regimes.

Findings

Disorder is irrelevant for  < 1/2 with summable correlations.

Disorder is relevant for  > 1/2, extending the Harris criterion.

In the case   (1,2), disorder relevance is proven for  > 1/.

Abstract

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $\alpha$ > 0, when the correlated sequence is given by another independent renewal set with loop exponent $\alpha$ > 0. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case $\alpha$ > 2 (summable correlations), disorder is irrelevant if $\alpha$ < 1/2 and relevant if $\alpha$ > 1/2, which extends the Harris criterion for independent disorder. The case $\alpha$ $\in$ (1, 2) (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha$ > 1/ $\alpha$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\alpha$ $\in$ (0, 1) is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.