License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.24
URN: urn:nbn:de:0030-drops-171463
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17146/
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Blanca, Antonio ; Gheissari, Reza

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

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LIPIcs-APPROX24.pdf (0.8 MB)

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.

BibTeX - Entry

@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2022.24,
  author =	{Blanca, Antonio and Gheissari, Reza},
  title =	{{Sampling from Potts on Random Graphs of Unbounded Degree via Random-Cluster Dynamics}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{24:1--24:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/17146},
  URN =		{urn:nbn:de:0030-drops-171463},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.24},
  annote =	{Keywords: Potts model, random-cluster model, random graphs, Markov chains, mixing time, tree uniqueness}
}

Keywords: Potts model, random-cluster model, random graphs, Markov chains, mixing time, tree uniqueness
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)
Issue Date: 2022
Date of publication: 15.09.2022

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