Rapid Mixing at the Uniqueness Threshold
🤖 AI Summary
This work resolves a long-standing open problem concerning the mixing time of Gibbs sampling at the computational phase transition point—the tree uniqueness threshold—for hardcore and Ising models. Specifically, it establishes the first rigorous polynomial upper and lower bounds on the mixing time of Glauber dynamics at criticality: for the hardcore model with fugacity λ = λ_c(Δ) and the Ising model with inverse temperature β = β_c on graphs of maximum degree Δ ≥ 3. The approach introduces a novel framework integrating field dynamics, proximal samplers, and localization analysis (Chen–Eldan framework), combined with spectral independence and high-dimensional probability tools. Results show that the hardcore model mixes in Õ(n^{2+4/e})–Ω(n^{4/3}) steps, while the Ising model mixes in Õ(n²)–Ω(n^{3/2}) steps at critical temperature, and achieves a tight bound of Õ(n^{3/2}) under the critical interaction matrix. This fills a fundamental gap in computational phase transition theory regarding convergence rates precisely at criticality.
Application Category
📝 Abstract
Over the past decades, a fascinating computational phase transition has been identified in sampling from Gibbs distributions. Specifically, for the hardcore model on graphs with n vertices and maximum degree Δ, the computational complexity of sampling from the Gibbs distribution, defined over the independent sets of the graph with vertex-weight λ>0, undergoes a sharp transition at the critical threshold λc(Δ) := (Δ−1)Δ−1/(Δ−2)Δ, known as the tree-uniqueness threshold: In the uniqueness regime where λλc(Δ), the Glauber dynamics exhibits exponential mixing time; furthermore, the sampling problem becomes intractable unless RP=NP. The computational complexity at the critical point λ = λc(Δ) 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≥ 3 at the uniqueness threshold λ = λc(Δ), the mixing time of Glauber dynamics is upper bounded by a polynomial in n, but is not nearly linear in the worst case: specifically, it falls between Õ(n(2+4e)+O(1/Δ)) and Ω(n4/3). For the Ising model (either antiferromagnetic or ferromagnetic), we establish similar results. For the Ising model on graphs with maximum degree Δ≥ 3 at the critical temperature β where |β| = βc(Δ), with the tree-uniqueness threshold βc(Δ) defined by (Δ−1)tanhβc(Δ)=1, we show that the mixing time of Glauber dynamics is upper bounded by Õ(n2 + O(1/Δ)) and lower bounded by Ω(n3/2) in the worst case. For the Ising model specified by a critical interaction matrix J with ∥ J ∥2=1, we obtain an upper bound Õ(n3/2) for the mixing time, matching the lower bound Ω(n3/2) on the complete graph up to a logarithmic factor. Our mixing time upper bounds hold regardless of whether the maximum degree Δ is constant. These bounds are derived from a new interpretation and analysis of the localization scheme method introduced by Chen and Eldan, 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.
Problem
Research questions and friction points this paper is trying to address.
Resolve computational complexity at critical phase transition threshold
Analyze mixing time of Glauber dynamics for hardcore model
Establish bounds for spectral independence at critical point
Innovation
Methods, ideas, or system contributions that make the work stand out.
Glauber dynamics mixing time analysis
Localization scheme method application
Spectral independence bounds at criticality
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