Average persistence in random walks

TL;DR

This paper investigates the average persistence probability of a random walk's expectation value in both homogeneous and disordered environments, revealing a logarithmic decay characterized by a new exponent and establishing a connection to the transverse-field Ising chain.

Contribution

It introduces a novel analysis of average persistence in random walks, deriving an exact logarithmic decay exponent and linking it to quantum spin chain dynamics.

Findings

Average persistence decays as (ln t)^theta' with theta' ≈ 0.191

Exact computation of average persistence for moderate system sizes

Established correspondence with magnetization autocorrelation in Ising chains

Abstract

We study the first passage time properties of an integrated Brownian curve both in homogeneous and disordered environments. In a disordered medium we relate the scaling properties of this center of mass persistence of a random walker to the average persistence, the latter being the probability P_pr(t) that the expectation value <x(t)> of the walker's position after time t has not returned to the initial value. The average persistence is then connected to the statistics of extreme events of homogeneous random walks which can be computed exactly for moderate system sizes. As a result we obtain a logarithmic dependence P_pr(t)~{ln(t)}^theta' with a new exponent theta'=0.191+/-0.002. We note on a complete correspondence between the average persistence of random walks and the magnetization autocorrelation function of the transverse-field Ising chain, in the homogeneous and disordered case.