One-arm exponents of high-dimensional percolation revisited

TL;DR

This paper provides a simplified proof for the one-arm probability estimates in high-dimensional percolation, improving existing techniques by leveraging recent correlation length estimates.

Contribution

It introduces a concise proof method for one-arm exponents in high-dimensional percolation, building on recent advances and simplifying prior complex techniques.

Findings

Established up-to-constants estimates for one-arm probabilities in high dimensions

Provided a simplified proof approach improving upon previous entropic techniques

Connected percolation results with high-dimensional Ising model studies

Abstract

We consider sufficiently spread-out Bernoulli percolation in dimensions ${d>6}$. We present a short and simple proof of the up-to-constants estimate for the one-arm probability in both the full-space and half-space settings. These results were previously established by Kozma and Nachmias and by Chatterjee and Hanson, respectively. Our proof improves upon the entropic technique introduced by Dewan and Muirhead, relying on a sharp estimate on a suitably chosen correlation length recently obtained by Duminil-Copin and Panis. This approach is inspired by our companion work, where we compute the one-arm exponent for several percolation models related to the high-dimensional Ising model.