A connection between the random pinning model and random walks in sparse random environments

TL;DR

This paper establishes a link between a one-dimensional random walk in a sparse environment and the random pinning model, revealing insights into return times and integrability properties in these stochastic systems.

Contribution

It introduces a novel connection between the random pinning model and sparse random walks, providing new analytical tools for their study.

Findings

Partition function equals mean return count in the sparse environment

Insights into integrability of return times in annealed setups

Enhanced understanding of the relationship between pinning models and random walks

Abstract

The purpose of this short note is to establish a connection between a one-dimensional random walk in a random sparse environment and the random pinning model. We show that the grand canonical partition function of the pinning model coincides with the mean number of returns to the origin for a random walk in a random sparse environment averaged on the randomness location. We obtain thereof some information on the integrability of the number of return times in the annealed and partially annealed setups.