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

Authors Antonio Blanca, Reza Gheissari

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Author Details

Antonio Blanca
  • Department of CSE, Pennsylvania State University, University Park, PA, USA
Reza Gheissari
  • Department of Statistics and EECS, University of California, Berkeley, CA, USA

Funding

  • Gheissari, Reza: Thanks the Miller Institute for Basic Research for its support.

Acknowledgements

The authors thank the anonymous referees for their helpful comments.

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Antonio Blanca and Reza Gheissari. Sampling from Potts on Random Graphs of Unbounded Degree via Random-Cluster Dynamics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.24

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(nlog 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 < p_u(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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Generating random combinatorial structures
  • Theory of computation → Random network models
  • Mathematics of computing → Probabilistic algorithms
  • Mathematics of computing → Markov processes
Keywords
  • Potts model
  • random-cluster model
  • random graphs
  • Markov chains
  • mixing time
  • tree uniqueness

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