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

Abstract

Consider critical site percolation on ${\mathbb{Z}}^d$ with $d\ge 2$. 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 $c{n}^{-(d^2+4d-2)}$ (with $c>0$ a constant which depends on $d$), thus giving an upper bound $d^2+4d-2$ for the above mentioned exponent.

10 pages, 1 figure