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 $\Delta\ge 3$ at the uniqueness threshold $\lambda = \lambda_c(\Delta)$, 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 $\Delta\ge 3$ at the critical temperature $\beta$ where $|\beta| = \beta_c(\Delta)$, with the tree-uniqueness threshold $\beta_c(\Delta)$, we show that the mixing time of Glauber dynamics is upper bounded by $\tilde{O}\left(n^{2 + O(1/\Delta)}\right)$ and lower bounded by $\Omega\left(n^{3/2}\right)$ 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 $\Omega\left(n^{3/2}\right)$ 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.

翻译：过去数十年间，在从吉布斯分布中采样的研究中，人们发现了一种引人入胜的计算相变现象。然而，在临界点处的计算复杂性仍然知之甚少，因为先前的算法结果与困难性结果都需要偏离该阈值一个常数松弛量。在本文中，我们解决了这一在临界相变阈值处的开放性问题，从而完善了计算相变的整体图景。我们证明，对于最大度为 $\Delta\ge 3$ 的图上的硬核模型，在唯一性阈值 $\lambda = \lambda_c(\Delta)$ 处，Glauber 动力学的混合时间上界为 $n$ 的多项式，但在最坏情况下并非近似线性。对于伊辛模型（无论是反铁磁还是铁磁），我们建立了类似的结果。对于最大度为 $\Delta\ge 3$ 的图上的伊辛模型，在临界温度 $\beta$ 处（其中 $|\beta| = \beta_c(\Delta)$，$\beta_c(\Delta)$ 为树唯一性阈值），我们证明 Glauber 动力学的混合时间在最坏情况下上界为 $\tilde{O}\left(n^{2 + O(1/\Delta)}\right)$，下界为 $\Omega\left(n^{3/2}\right)$。对于由临界相互作用矩阵 $J$（满足 $\left \lVert J \right \rVert_2=1$）指定的伊辛模型，我们得到了混合时间的上界 $\tilde{O}(n^{3/2})$，这与完全图上的下界 $\Omega\left(n^{3/2}\right)$ 在对数因子内匹配。我们的混合时间上界源自对 Chen 和 Eldan (2022) 提出的定位方案方法的一种新解释和分析，该方法被应用于硬核模型的场动力学和伊辛模型的近端采样器。作为我们上界和下界证明的关键步骤，我们针对最坏情况实例，在临界点处建立了谱独立性的亚线性上界和下界。