Convergence of $k$-point functions in high dimensional percolation

Abstract

We establish first-order asymptotics for the $k$-point connectivity function in critical Bernoulli percolation on ${\mathbb{Z}}^d$ assuming $d>6$ and Hara's two-point asymptotic. For every $k\ge 3$ and distinct macroscopically separated locations, the probability that the corresponding lattice points lie in the same open cluster, rescaled by $n^{(4-d)(k-1)-2}$, converges as $n\to \infty$ to an explicit sum of integrals indexed by trees. This confirms a conjecture of Aizenman and Newman. We also establish convergence of chemical-distance-decorated $k$-point functions. Conditionally on a set of $k$ point lying in the same cluster, this identifies the joint limiting distribution of all pairwise chemical distances after quadratic rescaling. It is described by binary trees with explicit laws along the edges. The decorated expansion also gives $k$ point asymptotics in the near-critical window. Our proof uses our previous results on convergence to the incipient infinite clusters to decouple macroscopic connections, as well as distributional convergence of the chemical distance.

53 pages. New results added and several mistakes corrected