Renewal convergence rates and correlation decay for homogeneous pinning models

TL;DR

This paper investigates the relationship between correlation decay and renewal convergence rates in homogeneous pinning models, revealing that near criticality these rates coincide, and providing sharp asymptotic behavior under certain conditions.

Contribution

It establishes the equivalence of correlation decay and renewal convergence rates near criticality and derives a local limit theorem for sharper asymptotics.

Findings

Correlation decay rate matches renewal convergence rate near criticality.

Away from criticality, these rates generally differ.

A local limit theorem describes the sharp asymptotic behavior of correlations.

Abstract

A class of discrete renewal processes with super-exponentially decaying inter-arrival distributions coincides with the infinite volume limit of general homogeneous pinning models in their localized phase. Pinning models are statistical mechanics systems to which a lot of attention has been devoted both for their relevance for applications and because they are solvable models exhibiting a non-trivial phase transition. The spatial decay of correlations in these systems is directly mapped to the speed of convergence to equilibrium for the associated renewal processes. We show that close to criticality, under general assumptions, the correlation decay rate, or the renewal convergence rate, coincides with the inter-arrival decay rate. We also show that, in general, this is false away from criticality. Under a stronger assumption on the inter-arrival distribution we establish a local limit theorem, capturing thus the sharp asymptotic behavior of correlations.