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On the Mixing Time of Glauber Dynamics for the Hard-Core and Related Models on G(n,d/n)

Authors Charilaos Efthymiou, Weiming Feng

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Author Details

Charilaos Efthymiou
  • Computer Science, University of Warwick, Coventry, UK
Weiming Feng
  • School of Informatics, University of Edinburgh, Edinburgh, UK

Funding

  • Efthymiou, Charilaos: EPSRC New Investigator Award (grant no. EP/V050842/1) and Centre of Discrete Mathematics and Applications (DIMAP), The University of Warwick.
  • Feng, Weiming: European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 947778).

Acknowledgements

Weiming Feng would like to thank Heng Guo for the helpful discussions.

Cite AsGet BibTex

Charilaos Efthymiou and Weiming Feng. On the Mixing Time of Glauber Dynamics for the Hard-Core and Related Models on G(n,d/n). In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.54

Abstract

We study the single-site Glauber dynamics for the fugacity λ, Hard-Core model on the random graph G(n, d/n). We show that for the typical instances of the random graph G(n,d/n) and for fugacity λ < {d^d} / {(d-1)^(d+1)}, the mixing time of Glauber dynamics is n^{1 + O(1/log log n)}. Our result improves on the recent elegant algorithm in [Bezáková, Galanis, Goldberg and Štefankovič; ICALP'22]. The algorithm there is an MCMC-based sampling algorithm, but it is not the Glauber dynamics. Our algorithm here is simpler, as we use the classic Glauber dynamics. Furthermore, the bounds on mixing time we prove are smaller than those in Bezáková et al. paper, hence our algorithm is also faster. The main challenge in our proof is handling vertices with unbounded degrees. We provide stronger results with regard the spectral independence via branching values and show that the our Gibbs distributions satisfy the approximate tensorisation of the entropy. We conjecture that the bounds we have here are optimal for G(n,d/n). As corollary of our analysis for the Hard-Core model, we also get bounds on the mixing time of the Glauber dynamics for the Monomer-Dimer model on G(n,d/n). The bounds we get for this model are slightly better than those we have for the Hard-Core model

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Discrete mathematics
Keywords
  • spin-system
  • spin-glass
  • sparse random (hyper)graph
  • approximate sampling
  • efficient algorithm

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