Improved Mixing of Critical Hardcore Model
Zongchen Chen, Tianhui Jiang
TL;DR
This work addresses sampling from the hardcore model on graphs with maximum degree $\Delta\ge3$ at the critical fugacity $\lambda_c(\Delta)$. It develops an optimal $\ell_\infty$-spectral-independence bound in the subcritical regime, and uses spectral-independence localization to derive a polynomial-time mixing bound for Glauber dynamics at criticality, achieving $T_{\mathrm{mix}}=\tilde{O}(n^{4+O(1/\Delta)})$. The main technical contribution is the explicit upper bound $C^\infty_{SI}=(1+\hat{x})/(1-d\hat{x})$ for the subcritical hardcore model, with $\hat{x}$ the fixed point of $F_{d,\lambda}$, plus a matching lower bound on finite trees that establishes tightness. This yields a substantial improvement over prior bounds (e.g., $\tilde{O}(n^{12.88+O(1/\Delta)})$) and demonstrates the power of the spectral-independence approach in handling critical-phase sampling problems.
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 $\tilde{O}(n^{4+O(1/Δ)})$ time on any $n$-vertex graph of maximum degree $Δ\geq3$, significantly improving the previous upper bound $\tilde{O}(n^{12.88+O(1/Δ)})$ by the recent work arXiv:2411.03413. Our improvement comes from an optimal bound on the $\ell_\infty$-spectral independence for the hardcore model at all subcritical fugacity $λ< λ_c(Δ)$.