2026-07-26T05:03:03Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:25098 2026-02-09T08:06:09Z ddc:004 open_access

Cutoff for the Swendsen–Wang Dynamics on the Complete Graph Blanca, Antonio Song, Zhezheng Markov chains mixing times cutoff phenomenon Potts model mean-field 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Antonio Blanca and Zhezheng Song 2025 Is Part Of LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FSTTCS.2025.17 urn:nbn:de:0030-drops-250987 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.17 eng https://creativecommons.org/licenses/by/4.0/legalcode