Yasodharan, S and Sundaresan, R (2023) Large-Time behaviour and the second eigenvalue problem for finite-state mean-field interacting particle systems. In: Advances in Applied Probability, 55 (1). 85 – 125.
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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 for a suitable constant 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. 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. © The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust.
| Item Type: | Journal Article |
|---|---|
| Publication: | Advances in Applied Probability |
| Publisher: | Cambridge University Press |
| Additional Information: | The copyright of this article belongs to the Authors. |
| Keywords: | Eigenvalues and eigenfunctions; Vlasov equation; Cycle; Empirical measure; Exit from a domain; Finite-state; Invariant measure; Large time behavior; McKean-Vlasov equations; Mean-field; Mean-field interactions; Metastabilities; Simulated annealing |
| Department/Centre: | Division of Electrical Sciences > Electrical Communication Engineering |
| Date Deposited: | 10 Mar 2023 10:38 |
| Last Modified: | 10 Mar 2023 10:38 |
| URI: | https://eprints.iisc.ac.in/id/eprint/80954 |
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