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Graph-combinatorial approach for large deviations of Markov chains

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Original languageEnglish
Article number295001
JournalJournal of Physics A
Volume55
Issue number29
DOIs
Accepted/In press17 Jun 2022
Published4 Jul 2022

Bibliographical note

Funding Information: GC and FC are thankful to Gianmichele Di Matteo for insightful discussions and to Mayank Shreshtha for having designed figure . FC is grateful to Hugo Touchette for pointing to interesting literature in the topic and for the hospitality in Stellenbosch (South Africa) during the writing stage of the manuscript. GC is supported by the EPSRC Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES, EP/L015854/1). Publisher Copyright: © 2022 The Author(s). Published by IOP Publishing Ltd.

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Abstract

We consider discrete-time Markov chains and study large deviations of the pair empirical occupation measure, which is useful to compute fluctuations of pure-additive and jump-type observables. We provide an exact expression for the finite-time moment generating function, which is split in cycles and paths contributions, and scaled cumulant generating function of the pair empirical occupation measure via a graph-combinatorial approach. The expression obtained allows us to give a physical interpretation of interaction and entropic terms, and of the Lagrange multipliers, and may serve as a starting point for sub-leading asymptotics. We illustrate the use of the method for a simple two-state Markov chain.

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