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.
From: Philippe Sosoe [view email]
[v1]
Sun, 7 Sep 2025 22:52:23 UTC (49 KB)
[v2]
Tue, 11 Nov 2025 20:42:40 UTC (54 KB)
[v3]
Sun, 30 Aug 2026 09:47:46 UTC (63 KB)
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