Arm exponents in high dimensional percolation

TL;DR

This paper investigates arm event probabilities in high-dimensional critical percolation, establishing decay rates of 1/r^2 for single arms and 1/r^{2k} for multiple disjoint arms, revealing precise asymptotic behaviors.

Contribution

It provides rigorous proofs of decay rates for arm probabilities in high-dimensional percolation, extending understanding beyond low-dimensional cases.

Findings

Probability of a single arm decays as 1/r^2

Probability of k disjoint arms decays as 1/r^{2k}

Results hold for large or sufficiently spread out lattices in high dimensions

Abstract

We study the probability that the origin is connected to the sphere of radius r (an arm event) in critical percolation in high dimensions, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We prove that this probability decays like 1/r^2. Furthermore, we show that the probability of having k disjoint arms to distance r emanating from the vicinity of the origin is 1/r^2k.