Limiting distribution of the chemical distance in high dimensional critical percolation

Abstract

We identify the asymptotic distribution of the chemical distance and other natural metrics (including the resistance) in high-dimensional critical percolation. When rescaled by the square of the Euclidean distance, each of these metrics converges in distribution to a multiple of the hitting time $T$ of a Brownian motion to hit $\mathbf{e}_1$ conditional on $\{T < \infty\}$. We extend these results to the near-critical regime, where the limiting distribution is now the analogue of $T$ for a killed Brownian motion. These results are intended to be the foundation for an understanding of the metric space structure of high-dimensional clusters. They follow from a general theorem we find interesting in its own right, a ``law of large numbers'' for local functions summed along the backbone of a long open connection. A mixing result for open clusters \cite{CCHS} in the form of a robust convergence to the incipient infinite cluster measure plays a key role in the proofs.