Oscillatory correlation of delayed random walks

TL;DR

This paper analyzes a delayed random walk model that exhibits oscillatory correlation behavior, providing analytical insights into its transient and stationary states, and relating it to Langevin equations with delay.

Contribution

It introduces a tractable model of delayed random walks with oscillatory correlations and explores its analytical and numerical properties, including connection to Langevin equations with delay.

Findings

The model exhibits oscillatory correlation functions both transiently and at stationarity.

Analytical solutions for the correlation functions are derived.

The model's behavior is related to Langevin equations with delay.

Abstract

We investigate analytically and numerically the statistical properties of a random walk model with delayed transition probability dependence (delayed random walk). The characteristic feature of such a model is the oscillatory behavior of its correlation function. We investigate a model whose transient and stationary oscillatory behavior is analytically tractable. The correspondence of the model with a Langevin equation with delay is also considered.