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

Authors Antonio Blanca, Xusheng Zhang



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Author Details

Antonio Blanca
  • Pennsylvania State University, University Park, PA, USA
Xusheng Zhang
  • Pennsylvania State University, University Park, PA, USA

Acknowledgements

The authors thank the anonymous referees for their constructive comments during the manuscript review process.

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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)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.53

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Gibbs sampling
Keywords
  • Markov chains
  • Spectral Independence
  • Potts model
  • Mixing Time

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