Rapid Mixing at the Uniqueness Threshold

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.