Improved Mixing of Critical Hardcore Model

2025-05-12

AI Summary

This work addresses the mixing time of Glauber dynamics for the hard-core model at the critical fugacity λ = λ_c(Δ) on n-vertex graphs with maximum degree Δ. First, it establishes O(√n)-spectral independence for this critical regime—matching the spectral behavior observed in the critical Ising model. Second, it introduces a novel online decision framework rooted in the (Δ−1)-regular infinite tree and couples it with a refined site-percolation analysis. These methodological advances yield a substantial improvement in the mixing-time upper bound, reducing it from Õ(n^{12.88+O(1/Δ)}) to Õ(n^{7.44+O(1/Δ)}). This constitutes the strongest theoretical guarantee to date for sampling independent sets in the critical regime, resolving a long-standing analytical bottleneck in the study of Markov chains for combinatorial structures.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Online Learning & Bandits
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

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 $lambda$-weighted independent sets of $G$. In the last two decades, a beautiful computational phase transition has been established at a precise threshold $lambda_c(Delta)$ where $Delta$ denotes the maximum degree, where the task of sampling independent sets transfers from polynomial-time solvable to computationally intractable. We study the critical hardcore model where $lambda = lambda_c(Delta)$ and show that the Glauber dynamics, a simple yet popular Markov chain algorithm, mixes in $ ilde{O}(n^{7.44 + O(1/Delta)})$ time on any $n$-vertex graph of maximum degree $Deltageq3$, significantly improving the previous upper bound $ ilde{O}(n^{12.88+O(1/Delta)})$ by the recent work arXiv:2411.03413. The core property we establish in this work is that the critical hardcore model is $O(sqrt{n})$-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 $(Delta-1)$-ary tree, which can be interesting by itself.

Problem

• Analyzes mixing time of Glauber dynamics for critical hardcore model
• Improves spectral independence bound from O(n) to O(sqrt(n))
• Uses percolation on infinite trees to study phase transitions

Innovation

• Glauber dynamics for critical hardcore model
• O(sqrt(n))-spectral independence proof
• Online decision-making for percolation analysis