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RANDOM
Rapid Mixing of Global Markov Chains via Spectral Independence: The Unbounded Degree Case

Authors: Antonio Blanca and Xusheng Zhang

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)


Abstract
We consider spin systems on general n-vertex graphs of unbounded degree and explore the effects of spectral independence on the rate of convergence to equilibrium of global Markov chains. Spectral independence is a novel way of quantifying the decay of correlations in spin system models, which has significantly advanced the study of Markov chains for spin systems. We prove that whenever spectral independence holds, the popular Swendsen-Wang dynamics for the q-state ferromagnetic Potts model on graphs of maximum degree Δ, where Δ is allowed to grow with n, converges in O((Δ log n)^c) steps where c > 0 is a constant independent of Δ and n. We also show a similar mixing time bound for the block dynamics of general spin systems, again assuming that spectral independence holds. Finally, for monotone spin systems such as the Ising model and the hardcore model on bipartite graphs, we show that spectral independence implies that the mixing time of the systematic scan dynamics is O(Δ^c log n) for a constant c > 0 independent of Δ and n. Systematic scan dynamics are widely popular but are notoriously difficult to analyze. This result implies optimal O(log n) mixing time bounds for any systematic scan dynamics of the ferromagnetic Ising model on general graphs up to the tree uniqueness threshold. Our main technical contribution is an improved factorization of the entropy functional: this is the common starting point for all our proofs. Specifically, we establish the so-called k-partite factorization of entropy with a constant that depends polynomially on the maximum degree of the graph.

Cite as

Antonio Blanca and Xusheng Zhang. Rapid Mixing of Global Markov Chains via Spectral Independence: The Unbounded Degree Case. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 53:1-53:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2023.53,
  author =	{Blanca, Antonio and Zhang, Xusheng},
  title =	{{Rapid Mixing of Global Markov Chains via Spectral Independence: The Unbounded Degree Case}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{53:1--53:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.53},
  URN =		{urn:nbn:de:0030-drops-188789},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.53},
  annote =	{Keywords: Markov chains, Spectral Independence, Potts model, Mixing Time}
}
Document
RANDOM
The Critical Mean-Field Chayes-Machta Dynamics

Authors: Antonio Blanca, Alistair Sinclair, and Xusheng Zhang

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)


Abstract
The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on n vertices. The random-cluster model is parametrized by an edge probability p and a cluster weight q. Our focus is on the critical regime: p = p_c(q) and q ∈ (1,2), where p_c(q) is the threshold corresponding to the order-disorder phase transition of the model. We show that the mixing time of the Chayes-Machta dynamics is O(log n ⋅ log log n) in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the log log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the Chayes-Machta dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting.

Cite as

Antonio Blanca, Alistair Sinclair, and Xusheng Zhang. The Critical Mean-Field Chayes-Machta Dynamics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 47:1-47:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2021.47,
  author =	{Blanca, Antonio and Sinclair, Alistair and Zhang, Xusheng},
  title =	{{The Critical Mean-Field Chayes-Machta Dynamics}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{47:1--47:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.47},
  URN =		{urn:nbn:de:0030-drops-147408},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.47},
  annote =	{Keywords: Markov Chains, Mixing Times, Random-cluster Model, Ising and Potts Models, Mean-field, Chayes-Machta Dynamics, Random Graphs}
}
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