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Large-time behaviour and the second eigenvalue problem for finite-state mean-field interacting particle systems

Part of: Computer system organization Special processes General theory of linear operators Limit theorems

Published online by Cambridge University Press:  08 July 2022

Sarath Yasodharan  and
Rajesh Sundaresan
Sarath Yasodharan*
Affiliation:
Indian Institute of Science
Rajesh Sundaresan*
Affiliation:
Indian Institute of Science
*
*Postal address: Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru 560012, India.
*Postal address: Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru 560012, India.
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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) of 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 $\exp\{N\Lambda\}$ for a suitable constant $\Lambda > 0$ , 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.

Keywords

Mean-field interactionMcKean–Vlasov equationmetastabilityexit from a domaincyclessimulated annealing

MSC classification

Primary: 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 47A75: Eigenvalue problems 68M20: Performance evaluation; queueing; scheduling
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 85 - 125
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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