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Cutoff for the Swendsen–Wang Dynamics on the Complete Graph

Authors Antonio Blanca, Zhezheng Song

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ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Probabilistic algorithms
  • Mathematics of computing → Markov processes
  • Mathematics of computing → Random graphs
Keywords
  • Markov chains
  • mixing times
  • cutoff phenomenon
  • Potts model
  • mean-field

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Abstract

We study the speed of convergence of the Swendsen-Wang (SW) dynamics for the q-state ferromagnetic Potts model on the n-vertex complete graph, known as the mean-field model. The SW dynamics was introduced as an attractive alternative to the local Glauber dynamics, often offering faster convergence rates to stationarity in a variety of settings. A series of works have characterized the asymptotic behavior of the speed of convergence of the mean-field SW dynamics for all q ≥ 2 and all values of the inverse temperature parameter β > 0. In particular, it is known that when β > q the mixing time of the SW dynamics is Θ(log n). We strengthen this result by showing that for all β > q, there exists a constant c(β,q) > 0 such that the mixing time of the SW dynamics is c(β,q) log n + Θ(1). This implies that the mean-field SW dynamics exhibits the cutoff phenomenon in this temperature regime, demonstrating that this Markov chain undergoes a sharp transition from "far from stationarity" to "well-mixed" within a narrow Θ(1) time window. The presence of cutoff is algorithmically significant, as simulating the chain for fewer steps than its mixing time could lead to highly biased samples.

Cite As Get BibTex

Antonio Blanca and Zhezheng Song. Cutoff for the Swendsen–Wang Dynamics on the Complete Graph. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.17

Author Details

Antonio Blanca
  • Department of Computer Science and Engineering, Pennsylvania State University, University Park, PA, USA
Zhezheng Song
  • Department of Computer Science and Engineering, Pennsylvania State University, University Park, PA, USA

Funding

  • Blanca, Antonio: Research supported in part by NSF CAREER grant 2143762.

Acknowledgements

This work transpired from a discussion between the first author, Andreas Galanis, Reza Gheissari, Eric Vigoda, and Daniel Štefankovič during the Workshop on Algorithms and Randomness at Georgia Tech in 2018 in which the high-level idea of the proof was outlined.

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