Sharp bounds on the half-space two-point function for high-dimensional Bernoulli percolation

TL;DR

This paper establishes sharp bounds on the critical two-point function in high-dimensional Bernoulli percolation within a half-space, advancing understanding of spatial correlations in such models.

Contribution

It provides an up-to-constant estimate for the two-point function in a half-space for dimensions greater than six, completing prior research and answering a recent open question.

Findings

Established sharp bounds on the two-point function

Extended results to high-dimensional half-spaces

Resolved an open problem in percolation theory

Abstract

We consider Bernoulli percolation on $\mathbb Z^d$ with $d>6$. We prove an up-to-constant estimate for the critical two-point function restricted to a half-space. This completes previous results of Chatterjee and Hanson (Commun. Pure Appl. Math., 2021), and Chatterjee, Hanson, and Sosoe (Commun. Math. Phys., 2023), and solves a question asked by Hutchcroft, Michta, and Slade (Ann. Probab., 2023).