Limiting distribution of the chemical distance in high dimensional critical percolation

Abstract

We identify the asymptotic distribution of the chemical distance in high-dimensional critical Bernoulli percolation. Namely, we show that the distance between the origin and a distant vertex conditioned to lie in the cluster of the origin converges in distribution when rescaled by a multiple the square of the Euclidean distance. The limiting distribution has an explicit density and coincides with the distribution of the time for a Brownian motion in $\mathbb{R}^d$ conditioned to hit a given unit vector to reach its target. Our result follows from a general moment computation for quantities that have an additive structure across the pivotal edges on a long-range connection in percolation. In addition to the number of pivotal edges in a long connection, this also includes the effective resistance. The existence of the incipient infinite cluster limit, in a form recently established, plays a key role in the derivation of our results.