Random walks on $\mathbb{Z}$ with metastable Gaussian distribution caused by linear drift with application to the contact process on the complete graph

TL;DR

This paper analyzes random walks on integers with a linear drift towards zero, establishing their metastable Gaussian distribution and applying these results to the contact process on complete graphs to describe its metastable behavior.

Contribution

It provides a theoretical framework for metastability in random walks with linear drift and applies it to the contact process, demonstrating Gaussian convergence of the infected fraction.

Findings

Explicit bounds on the distribution's distance to Gaussian during metastability.

Application of results to the contact process showing Gaussian convergence.

Identification of fast-growing time windows for distribution approximation.

Abstract

We study random walks on $\mathbb{Z}$ which have a linear (or almost linear) drift towards 0 in a range around 0. This drift leads to a metastable Gaussian distribution centered at zero. We give specific, fast growing, time windows where we can explicitely bound the distance of the distribution of the walk to an appropriate Gaussian. In this way we give a solid theoretical foundation to the notion of metastability. We show that the supercritical contact process on the complete graph has a drift towards its equilibrium point which is locally linear and that our results for random walks apply. This leads to the conclusion that the infected fraction of the population in metastability (when properly scaled) converges in distribution to a Gaussian, uniformly for all times in a fast growing interval.