An upper bound on the two-arms exponent for critical percolation on $\mathbb{Z}^d$

TL;DR

This paper establishes an upper bound on the two-arms exponent in critical percolation on integer lattices, providing new insights into the decay rate of two-arms probabilities in high dimensions.

Contribution

It derives a novel upper bound for the two-arms exponent in critical percolation on $ abla^d$, filling a gap in the understanding of this exponent's behavior.

Findings

Two-arms probability decays at least as fast as $c n^{-(d^2 + 4 d -2)}$

Provides the first explicit upper bound for the two-arms exponent in high dimensions

Enhances understanding of critical percolation cluster connectivity properties.

Abstract

Consider critical site percolation on $\mathbb{Z}^d$ with $d \geq 2$. Cerf (2015) pointed out that from classical work by Aizenman, Kesten and Newman (1987) and Gandolfi, Grimmett and Russo (1988) one can obtain that the two-arms exponent is at least $1/2$. The paper by Cerf slightly improves that lower bound. Except for $d=2$ and for high $d$, no upper bound for this exponent seems to be known in the literature so far (not even implicity). We show that the distance-$n$ two-arms probability is at least $c n^{-(d^2 + 4 d -2)}$ (with $c >0$ a constant which depends on $d$), thus giving an upper bound $d^2 + 4 d -2$ for the above mentioned exponent.