Large Time Behaviour and the Second Eigenvalue Problem for Finite State Mean-Field Interacting Particle Systems

TL;DR

This paper investigates the long-term behavior of finite state mean-field interacting particle systems, providing exponential estimates for convergence times, asymptotics of the second eigenvalue, and applications to global minimum search via controlled particle addition.

Contribution

It offers new exponential estimates for convergence times and eigenvalue asymptotics, linking large deviation principles to the system's spectral properties and optimization applications.

Findings

Convergence time scales as exp(NΛ) for large N.

Second eigenvalue's absolute value scales as exp(-NΛ).

Empirical measure converges to a global minimum under controlled particle addition.

Abstract

This article examines large time behaviour of finite state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) on the time required for convergence of the empirical measure process of the $N$-particle system to its invariant measure; we show that when time is of the order of $\exp\{N\Lambda\}$ for a suitable constant $\Lambda \geq 0$, the process has mixed well and it is close to its invariant measure. We then obtain large-$N$ asymptotics of the second largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as $\exp\{-N\Lambda\}$. The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-$N$ limit. As an application of the study of large time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain `entropy' function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.