We prove tight bounds on the site percolation threshold for $k$-uniform hypergraphs of maximum degree $\Delta$ and for $k$-uniform hypergraphs of maximum degree $\Delta$ in which any pair of edges overlaps in at most $r$ vertices. The hypergraphs that achieve these bounds are hypertrees, but unlike in the case of graphs, there are many different $k$-uniform, $\Delta$-regular hypertrees. Determining the extremal tree for a given $k, \Delta, r$ involves an optimization problem, and our bounds arise from a convex relaxation of this problem. By combining our percolation bounds with the method of disagreement percolation we obtain improved bounds on the uniqueness threshold for the hard-core model on hypergraphs satisfying the same constraints. Our uniqueness conditions imply exponential weak spatial mixing, and go beyond the known bounds for rapid mixing of local Markov chains and existence of efficient approximate counting and sampling algorithms. Our results lead to natural conjectures regarding the aforementioned algorithmic tasks, based on the intuition that uniqueness thresholds for the extremal hypertrees for percolation determine computational thresholds.

翻译：我们证明了对最大度为$\Delta$的$k$-均匀超图以及任意两条边至多共享$r$个顶点的最大度为$\Delta$的$k$-均匀超图，其位点渗流阈值的紧界。达到这些界的超图是超树，但与图的情况不同，存在许多不同的$k$-均匀、$\Delta$-正则超树。确定给定$k,\Delta,r$下的极值超树涉及一个优化问题，而我们的界源自该问题的凸松弛。通过将渗流界与分歧渗流方法相结合，我们改进了满足相同约束的超图上硬核模型唯一性阈值的已知界。我们的唯一性条件蕴含指数级弱空间混合，并超越了局部马尔可夫链快速混合及有效近似计数与采样算法存在性的现有结论。基于极值渗流超树的唯一性阈值决定计算阈值的直觉，我们的结果为前述算法任务提出了自然猜想。