Sampling from Potts on Random Graphs of Unbounded Degree via Random-Cluster Dynamics

Antonio Blanca*, Reza Gheissari

*Corresponding author for this work

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

2 Scopus citations

Abstract

We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The random-cluster model is parametrized by an edge probability p ∈ (0, 1) and a cluster weight q > 0. We establish that for every q ≥ 1, the random-cluster Glauber dynamics mixes in optimal Θ(n log n) steps on n-vertex random graphs having a prescribed degree sequence with bounded average branching γ throughout the full high-temperature uniqueness regime p < pu(q, γ). The family of random graph models we consider includes the Erdős-Rényi random graph G(n, γ/n), and so we provide the first polynomial-time sampling algorithm for the ferromagnetic Potts model on Erdős-Rényi random graphs for the full tree uniqueness regime. We accompany our results with mixing time lower bounds (exponential in the largest degree) for the Potts Glauber dynamics, in the same settings where our Θ(n log n) bounds for the random-cluster Glauber dynamics apply. This reveals a novel and significant computational advantage of random-cluster based algorithms for sampling from the Potts model at high temperatures.

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022
EditorsAmit Chakrabarti, Chaitanya Swamy
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772495
DOIs
StatePublished - Sep 1 2022
Event25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022 - Virtual, Urbana-Champaign, United States
Duration: Sep 19 2022Sep 21 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume245
ISSN (Print)1868-8969

Conference

Conference25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022
Country/TerritoryUnited States
CityVirtual, Urbana-Champaign
Period9/19/229/21/22

Keywords

  • Markov chains
  • mixing time
  • Potts model
  • random graphs
  • random-cluster model
  • tree uniqueness

ASJC Scopus subject areas

  • Software

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