Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

TL;DR

This paper investigates high-dimensional percolation, establishing exponential decay of the one-arm probability ratio below criticality, and provides sharp estimates for critical quantities, advancing understanding of scaling and cluster behavior.

Contribution

It proves exponential decay of the one-arm probability ratio and offers sharp bounds for cluster volumes and distances at criticality, confirming aspects of scaling theory in high dimensions.

Findings

Exponential decay of the one-arm probability ratio below p_c.

Sharp bounds for cluster volume tail behavior.

Tightness of the number of spanning clusters at criticality.

Abstract

In high dimensional percolation at parameter $p < p_c$, the one-arm probability $\pi_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $\pi_p(n) / \pi_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at "mesoscopic distance" from the boundary of half-spaces. As a corollary, we obtain the tightness of the number of spanning clusters of a diameter $n$ box on scale $n^{d-6}$; this result complements a lower bound of Aizenman.