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

Cheliotis, Dimitris; Chino, Yuki; Poisat, Julien

doi:10.5802/ahl.11

Cheliotis, Dimitris; Chino, Yuki; Poisat, Julien

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

Annales Henri Lebesgue, Volume 2 (2019), pp. 281-329

Metadata

DOI10.5802/ahl.11

Keywords Pinning model , localization transition , free energy , correlated disorder , renewal , disorder relevance , Harris criterion , smoothing inequality , second moment.

Abstract

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $α > 0$ , when the correlated sequence is given by another independent renewal set with loop exponent $\hat{α} > 0$ . 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 $\hat{α} > 2$ (summable correlations), disorder is irrelevant if $α < 1 / 2$ and relevant if $α > 1 / 2$ , which extends the Harris criterion for independent disorder. The case $\hat{α} \in (1, 2)$ (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $α > 1 / \hat{α}$ , a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\hat{α} \in (0, 1)$ 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.

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