License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.528
URN: urn:nbn:de:0030-drops-53227
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5322/
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Blanca, Antonio ; Sinclair, Alistair

Dynamics for the Mean-field Random-cluster Model

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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 (lambda_s,lambda_S) of the model parameter lambda in which the dynamics undergoes an exponential slowdown. Namely, we prove that the mixing time is Theta(log n) if lambda is not in [lambda_s,lambda_S], and e^Omega(sqrt{n}) when lambda is in (lambda_s,lambda_S). 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 (lambda_s,lambda_S).

BibTeX - Entry

@InProceedings{blanca_et_al:LIPIcs:2015:5322,
  author =	{Antonio Blanca and Alistair Sinclair},
  title =	{{Dynamics for the Mean-field Random-cluster Model}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{528--543},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5322},
  URN =		{urn:nbn:de:0030-drops-53227},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.528},
  annote =	{Keywords: random-cluster model, random graphs, Markov chains, statistical physics, dynamics}
}

Keywords: random-cluster model, random graphs, Markov chains, statistical physics, dynamics
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)
Issue Date: 2015
Date of publication: 13.08.2015

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