On quenched and annealed critical curves of random pinning model with finite range correlations

TL;DR

This paper investigates the critical behavior of a directed polymer model with correlated disorder, deriving explicit formulas for the annealed critical curve and establishing conditions for disorder irrelevance using renewal theory and second moment methods.

Contribution

It generalizes the annealed critical curve calculation to correlated disorder modeled as a finite-range moving average, extending previous i.i.d. results.

Findings

Explicit annealed critical curve formulas for q=1, 2, and weak disorder asymptotics.

Disorder irrelevance established for certain correlated disorder cases.

Annealed and quenched critical curves coincide under specified conditions.

Abstract

This paper focuses on directed polymers pinned at a disordered and correlated interface. We assume that the disorder sequence is a q-order moving average and show that the critical curve of the annealed model can be expressed in terms of the Perron-Frobenius eigenvalue of an explicit transfer matrix, which generalizes the annealed bound of the critical curve for i.i.d. disorder. We provide explicit values of the annealed critical curve for $q=1$, 2 and a weak disorder asymptotic in the general case. Following the renewal theory approach of pinning, the processes arising in the study of the annealed model are particular Markov renewal processes. We consider the intersection of two replicas of this process to prove a result of disorder irrelevance (i.e. quenched and annealed critical curves as well as exponents coincide) via the method of second moment.