  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.
