Dynamics for the mean-field random-cluster model

Antonio Blanca, Alistair Sinclair

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

15 Scopus citations

Abstract

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and random spanning trees, but its dynamics have so far largely resisted analysis. In this paper we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, and identify a critical regime (λs, λS) of the model parameter λ in which the dynamics undergoes an exponential slowdown. Namely, we prove that the mixing time is θ(log n) if ∉ 62 [λs, λS], and exp((p n)) when λ ∈ (λs, λS). These results hold for all values of the second model parameter q > 1. In addition, we prove that the local heat-bath dynamics undergoes a similar exponential slowdown in (λs, λS).

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015
EditorsNaveen Garg, Klaus Jansen, Anup Rao, Jose D. P. Rolim
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages528-543
Number of pages16
ISBN (Electronic)9783939897897
DOIs
StatePublished - Aug 1 2015
Event18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 - Princeton, United States
Duration: Aug 24 2015Aug 26 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume40
ISSN (Print)1868-8969

Conference

Conference18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015
CountryUnited States
CityPrinceton
Period8/24/158/26/15

All Science Journal Classification (ASJC) codes

  • Software

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    Blanca, A., & Sinclair, A. (2015). Dynamics for the mean-field random-cluster model. In N. Garg, K. Jansen, A. Rao, & J. D. P. Rolim (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015 (pp. 528-543). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 40). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.528