On the critical curves of the Pinning and Copolymer models in Correlated Gaussian environment

TL;DR

This paper studies the effects of correlated Gaussian environments on disordered copolymer and pinning models, revealing how correlations influence critical points, exponents, and the weak coupling limit of critical curves.

Contribution

It extends the analysis of critical curves in disordered models to correlated environments, generalizing previous IID results and computing weak coupling limits with summable correlations.

Findings

Disorder relevance in non-negative correlated environments.

Annealed model becomes non-trivial with some negative correlations.

Explicit weak coupling limits of critical curves for finite mean return distributions.

Abstract

We investigate the disordered copolymer and pinning models, in the case of a correlated Gaussian environment with summable correlations, and when the return distribution of the underlying renewal process has a polynomial tail. As far as the copolymer model is concerned, we prove disorder relevance both in terms of critical points and critical exponents, in the case of non-negative correlations. When some of the correlations are negative, even the annealed model becomes non-trivial. Moreover, when the return distribution has a finite mean, we are able to compute the weak coupling limit of the critical curves for both models, with no restriction on the correlations other than summability. This generalizes the result of Berger, Caravenna, Poisat, Sun and Zygouras \cite{cf:BCPSZ} to the correlated case. Interestingly, in the copolymer model, the weak coupling limit of the critical curve turns out to be the maximum of two quantities: one generalizing the limit found in the IID case \cite{cf:BCPSZ}, the other one generalizing the so-called Monthus bound.