On the irrelevant disorder regime of pinning models

TL;DR

This paper advances understanding of the irrelevant disorder regime in polymer pinning models by using interpolation and replica coupling methods to derive precise free energy expansions and analyze correlation length exponents.

Contribution

It provides new rigorous results on the irrelevant disorder regime, including the first order free energy expansion and equality of correlation length exponents, using renewal theory techniques.

Findings

Computed the first order term in free energy expansion near criticality.

Showed quenched and quenched averaged correlation length exponents coincide.

Applied renewal theory to study intersection of renewal sequences.

Abstract

Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that is now rigorous. In this work, we exploit interpolation and replica coupling methods to obtain sharper results on the irrelevant disorder regime of pinning models. In particular, in this regime, we compute the first order term in the expansion of the free energy close to criticality and this term coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and quenched averaged correlation length exponents coincide, while, in general, they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals.