Mixing time power laws at criticality

Yun Long*, Asaf Nachmias, Yuval Peres

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations

Abstract

We study the mixing time of some Markov Chains converging to critical physical models. These models are indexed by a parameter β and there exists some critical value βc where the model undergoes a phase transition. According to Physics lore, the mixing time of such Markov Chains is often of logarithmic order outside the critical regime, when β ≠ βc, and satisfies some power law at criticality, when β= βc. We prove this in the two following settings: 1. Lazy random walk on the critical percolation cluster of "mean-field" graphs, which include the complete graph and random d-regular graphs. The critical mixing time here is of order Θ(n). This answers a question of Benjamini, Kozma and Wormald[4]. 2. Swendsen-Wang dynamics [33] on the complete graph. The critical mixing time here is of order Θ(n1/4). This improves results of Cooper, Dyer, Frieze and Rue [9]. In both settings, the main tool is understanding the Markov Chain dynamics via properties of critical percolation on the underlying graph.

Original languageEnglish
Title of host publicationProceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007
Pages205-214
Number of pages10
DOIs
StatePublished - 2007
Externally publishedYes
Event48th Annual Symposium on Foundations of Computer Science, FOCS 2007 - Providence, RI, United States
Duration: 20 Oct 200723 Oct 2007

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference48th Annual Symposium on Foundations of Computer Science, FOCS 2007
Country/TerritoryUnited States
CityProvidence, RI
Period20/10/0723/10/07

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