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Improved Mixing of Critical Hardcore Model

Authors Zongchen Chen , Tianhui Jiang

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ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Markov processes
Keywords
  • Hardcore model
  • Phase transition
  • Glauber dynamics
  • Spectral independence
  • Online decision making
  • Site percolation

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Abstract

The hardcore model is one of the most classic and widely studied examples of undirected graphical models. Given a graph G, the hardcore model describes a Gibbs distribution of λ-weighted independent sets of G. In the last two decades, a beautiful computational phase transition has been established at a precise threshold λ_c(Δ) where Δ denotes the maximum degree, where the task of sampling independent sets transitions from polynomial-time solvable to computationally intractable. We study the critical hardcore model where λ = λ_c(Δ) and show that the Glauber dynamics, a simple yet popular Markov chain algorithm, mixes in Õ(n^{7.44 + O(1/Δ)}) time on any n-vertex graph of maximum degree Δ ≥ 3, significantly improving the previous upper bound Õ(n^{12.88 + O(1/Δ)}) by the recent work [Chen et al., 2024]. The core property we establish in this work is that the critical hardcore model is O(√n)-spectrally independent, improving the trivial bound of n and matching the critical behavior of the Ising model. Our proof approach utilizes an online decision-making framework to study a site percolation model on the infinite (Δ-1)-ary tree, which can be interesting by itself.

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Zongchen Chen and Tianhui Jiang. Improved Mixing of Critical Hardcore Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 51:1-51:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.51

Author Details

Zongchen Chen
  • School of Computer Science, Georgia Institute of Technology, Atlanta, GA, USA
Tianhui Jiang
  • Zhiyuan College, Shanghai Jiao Tong University, China

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