Rapid Mixing of Global Markov Chains via Spectral Independence: The Unbounded Degree Case

Antonio Blanca, Xusheng Zhang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We consider spin systems on general n-vertex graphs of unbounded degree and explore the effects of spectral independence on the rate of convergence to equilibrium of global Markov chains. Spectral independence is a novel way of quantifying the decay of correlations in spin system models, which has significantly advanced the study of Markov chains for spin systems. We prove that whenever spectral independence holds, the popular Swendsen–Wang dynamics for the q-state ferromagnetic Potts model on graphs of maximum degree ∆, where ∆ is allowed to grow with n, converges in O((∆ log n)c) steps where c > 0 is a constant independent of ∆ and n. We also show a similar mixing time bound for the block dynamics of general spin systems, again assuming that spectral independence holds. Finally, for monotone spin systems such as the Ising model and the hardcore model on bipartite graphs, we show that spectral independence implies that the mixing time of the systematic scan dynamics is O(∆c log n) for a constant c > 0 independent of ∆ and n. Systematic scan dynamics are widely popular but are notoriously difficult to analyze. This result implies optimal O(log n) mixing time bounds for any systematic scan dynamics of the ferromagnetic Ising model on general graphs up to the tree uniqueness threshold. Our main technical contribution is an improved factorization of the entropy functional: this is the common starting point for all our proofs. Specifically, we establish the so-called k-partite factorization of entropy with a constant that depends polynomially on the maximum degree of the graph.

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023
EditorsNicole Megow, Adam Smith
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772969
DOIs
StatePublished - Sep 2023
Event26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023 - Atlanta, United States
Duration: Sep 11 2023Sep 13 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume275
ISSN (Print)1868-8969

Conference

Conference26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023
Country/TerritoryUnited States
CityAtlanta
Period9/11/239/13/23

All Science Journal Classification (ASJC) codes

  • Software

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