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

Sarath Yasodharan, Rajesh Sundaresan

Published 2019-09-09Version 1

This article examines large time behaviour and the second eigenvalue problem for Markovian mean-field interacting particle systems with jumps. Our first main result is on the time required for convergence of the empirical measure process of the particle system to its invariant measure; we show that, there is a constant $\Lambda \geq 0$ such that, when there are $N$ particles in the system and when time is of the order $\exp\{N(\Lambda+O(1))\}$, the process has mixed well and is very close to its invariant measure. We then obtain large-$N$ asymptotics of the second 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 like $\exp\{-N\Lambda\}$. 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.