Intersection exponents for biased random walks on discrete cylinders

TL;DR

This paper establishes the existence and real analyticity of intersection exponents for biased random walks on discrete cylinders, and proves exponential convergence to stationarity of an associated Markov chain.

Contribution

It introduces the first proof of intersection exponents for biased walks on cylinders and demonstrates their analytic dependence on parameters, along with convergence rates.

Findings

Existence of intersection exponents xi(k,lambda)

Real analyticity of these exponents as functions of lambda

Exponential convergence to stationarity of a related Markov chain

Abstract

We prove existence of intersection exponents xi(k,lambda) for biased random walks on d-dimensional half-infinite discrete cylinders, and show that, as functions of lambda, these exponents are real analytic. As part of the argument, we prove convergence to stationarity of a time-inhomogeneous Markov chain on half-infinite random paths. Furthermore, we show this convergence takes place at exponential rate, an estimate obtained via a coupling of weighted half-infinite paths.