Exponentially slow mixing in the mean-field Swendsen-Wang dynamics

Reza Gheissari, Eyal Lubetzky, Yuval Peres

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

Abstract

Swendsen-Wang dynamics for the Potts model was proposed in the late 1980's as an alternative to single-site heat-bath dynamics, in which global updates allow this MCMC sampler to switch between metastable states and ideally mix faster. Gore and Jerrum (1997) found that this dynamics may in fact exhibit slow mixing: they showed that, for the Potts model with q ≥ 3 colors on the complete graph on n vertices at the critical pointc(q), Swendsen-Wang dynamics has tmix ≥ exp(√c/n). Galanis et al. (2015) showed that tmix exp(cn1=3) throughout the critical window (βs; βS) around βc, and Blanca and Sinclair (2015) established that tmix ≥ exp(c p n) in the critical window for corresponding mean-field FK model, which implied the same bound for Swendsen-Wang via known comparison estimates. In both cases, an upper bound of tmix β exp(c0n) was known. Here we show that the mixing time is truly exponential in n: namely, tmix β exp(cn) for Swendsen-Wang dynamics when q ≥ 3 and β ϵ (βs; βS), and the same bound holds for the related MCMC samplers for the mean-field FK model when q > 2.

Original languageEnglish (US)
Title of host publication29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
EditorsArtur Czumaj
PublisherAssociation for Computing Machinery
Pages1981-1988
Number of pages8
ISBN (Electronic)9781611975031
DOIs
StatePublished - 2018
Event29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States
Duration: Jan 7 2018Jan 10 2018

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
CountryUnited States
CityNew Orleans
Period1/7/181/10/18

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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  • Cite this

    Gheissari, R., Lubetzky, E., & Peres, Y. (2018). Exponentially slow mixing in the mean-field Swendsen-Wang dynamics. In A. Czumaj (Ed.), 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 (pp. 1981-1988). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery. https://doi.org/10.1137/1.9781611975031.129