Rapid Mixing at the Uniqueness Threshold

TL;DR

This work resolves the long-standing question of Glauber-dynamics mixing at the uniqueness threshold for two-spin models, establishing polynomial upper bounds at criticality for the hardcore model and for Ising, with matching or near-matching lower bounds on worst-case instances. It introduces and exploits an information-theoretic reinterpretation of localization schemes (field dynamics and proximal samplers) to derive rapid mixing at criticality, along with sharp $O(\sqrt{n})$ spectral independence bounds for the critical Ising case with $\|J\|_2=1$. The analysis hinges on a unified down-up/continuous-time perspective, entropic stability, approximate tensorization of entropy, and anti-concentration results on random regular bipartite graphs to prove lower bounds. The paper further provides sub-exponential deterministic counting algorithms at criticality and discusses the broader implications for understanding phase transitions in computational complexity of sampling and counting. Overall, it completes the computational-phase-transition picture for these fundamental models and offers new tools for analyzing Markov chains at criticality.

Abstract

Over the past decades, a fascinating computational phase transition has been identified in sampling from Gibbs distributions. Though, the computational complexity at the critical point remains poorly understood, as previous algorithmic and hardness results all required a constant slack from this threshold. In this paper, we resolve this open question at the critical phase transition threshold, thus completing the picture of the computational phase transition. We show that for the hardcore model on graphs with maximum degree $Δ\ge 3$ at the uniqueness threshold $λ= λ_c(Δ)$, the mixing time of Glauber dynamics is upper bounded by a polynomial in $n$, but is not nearly linear in the worst case. For the Ising model (either antiferromagnetic or ferromagnetic), we establish similar results. For the Ising model on graphs with maximum degree $Δ\ge 3$ at the critical temperature $β$ where $|β| = β_c(Δ)$, with the tree-uniqueness threshold $β_c(Δ)$, we show that the mixing time of Glauber dynamics is upper bounded by $\tilde{O}\left(n^{3 + O(1/Δ)}\right)$ and lower bounded by $Ω\left(n^{3/2}\right)$ in the worst case. For the Ising model specified by a critical interaction matrix $J$ with $\left \lVert J \right \rVert_2=1$, we obtain an upper bound $\tilde{O}(n^{3/2})$ for the mixing time, matching the lower bound $Ω\left(n^{3/2}\right)$ on the complete graph up to a logarithmic factor. Our mixing time upper bounds are derived from a new interpretation and analysis of the localization scheme method introduced by Chen and Eldan (2022), applied to the field dynamics for the hardcore model and the proximal sampler for the Ising model. As key steps in both our upper and lower bounds, we establish sub-linear upper and lower bounds for spectral independence at the critical point for worst-case instances.

Figures (3)

• Figure 1: sub-critical
• Figure 2: critical
• Figure 3: super-critical

Theorems & Definitions (100)

• Theorem 1.1: Hardcore Model at the Critical Fugacity
• Theorem 1.2: Ising Model at the Critical Temperature
• Theorem 1.3: Ising Model at the Critical Interaction Norm
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Definition 2.1: Poincaré inequality and modified log-Sobolev inequality
• Definition 2.2: Approximate tensorization of variance/entropy caputo2015approximate
• Lemma 2.3: Folklore
• Definition 2.4: Spectral independence anari2020spectral