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Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point
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  • Published: 01 December 2010

Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point

  • Christian Borgs1,
  • Jennifer T. Chayes1 &
  • Prasad Tetali2 

Probability Theory and Related Fields volume 152, pages 509–557 (2012)Cite this article

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  • 29 Citations

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Abstract

We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice \({\mathbb{Z}^{d}}\)—heat bath dynamics and the Swendsen–Wang algorithm—and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the Swendsen–Wang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in L d-1. This work provides the first upper bound of this form for the Swendsen–Wang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L/(log L)2.

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Author information

Authors and Affiliations

  1. Microsoft Research, 1 Memorial Drive, Cambridge, MA, 02124, USA

    Christian Borgs & Jennifer T. Chayes

  2. School of Mathematics and School of Computer Science, Georgia Tech, Atlanta, GA, 30332-0160, USA

    Prasad Tetali

Authors
  1. Christian Borgs
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  2. Jennifer T. Chayes
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  3. Prasad Tetali
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Corresponding author

Correspondence to Prasad Tetali.

Additional information

P. Tetali research was supported in part by the NSF Grants DMS-9800351, DMS-0401239, DMS-0701043.

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Borgs, C., Chayes, J.T. & Tetali, P. Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point. Probab. Theory Relat. Fields 152, 509–557 (2012). https://doi.org/10.1007/s00440-010-0329-0

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  • Received: 23 August 2008

  • Revised: 18 October 2010

  • Published: 01 December 2010

  • Issue Date: April 2012

  • DOI: https://doi.org/10.1007/s00440-010-0329-0

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Keywords

  • Pirogov–Sinai theory
  • Contour representation
  • Heat bath
  • Partition width

Mathematics Subject Classification (2000)

  • Primary 60J10
  • Secondary 60K35
  • 68Q87
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