Thermalization And Convergence To Equilibrium Of The Noisy Voter Model

TL;DR

This paper analyzes the convergence behavior of the noisy voter model on complete graphs, revealing the absence of cut-off in convergence to equilibrium but the presence of a cut-off in thermalization, using Stein's method and PDE tools.

Contribution

It provides a detailed profile of convergence and thermalization in the noisy voter model, introducing new analytical techniques for studying Markov chain observables.

Findings

No cut-off in convergence to equilibrium under natural noise conditions.

Thermalization exhibits a cut-off profile.

Analytical approach using Stein's method and PDE theory.

Abstract

We investigate the convergence towards equilibrium of the noisy voter model, evolving in the complete graph with n vertices. The noisy voter model is a version of the voter model, on which individuals change their opinions randomly due to external noise. Specifically, we determine the profile of convergence, in Kantorovich distance (also known as 1-Wasserstein distance), which corresponds to the Kantorovich distance between the marginals of a Wright-Fisher diffusion and its stationary measure. In particular, we demonstrate that the model does not exhibit cut-off under natural noise intensity conditions. In addition, we study the time the model needs to forget the initial location of particles, which we interpret as the Kantorovich distance between the laws of the model with particles in fixed initial positions and in positions chosen uniformly at random. We call this process thermalization and we show that thermalization does exhibit a cut-off profile. Our approach relies on Stein's method and analytical tools from PDE theory, which may be of independent interest for the quantitative study of observables of Markov chains.