Metastability in loss networks with dynamic alternative routing

Olesker-Taylor, Sam

doi:10.1214/21-aap1711

Cited by 1 publication

(1 citation statement)

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“…This model arises in the context of dynamic alternate routing in loss networks. For certain values of , the limiting ODE (1.1) possesses two stable equilibria say and and an unstable equilibrium say [23, 38]. Thus, Assumption (B1) is satisfied with , , .…”

Section: Large-time Behaviourmentioning

confidence: 99%

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

Yasodharan

Sundaresan

2022

Adv. Appl. Probab.

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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.

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