Mean-field behavior for nearest-neighbor percolation in $d>10$

Abstract

We prove that nearest-neighbor percolation in dimensions $d\geq 11$ displays mean-field behavior by proving that the infrared bound holds, in turn implying the finiteness of the percolation triangle diagram. The finiteness of the triangle implies the existence and mean-field values of various critical exponents, such as $\gamma =1, \beta =1, \delta =2$. We also prove sharp $x$-space asymptotics for the two-point function and the existence of various arm exponents. Such results had previously been obtained in unpublished work by Hara and Slade for nearest-neighbor percolation in dimension $d\geq 19$, so that we bring the dimension above which mean-field behavior is rigorously proved down from $19$ to $11$. Our results also imply sharp bounds on the critical value of nearest-neighbor percolation on $\mathbb{Z} ^d$, which are provably at most $1.31\%$ off in $d=11$. We make use of the general method analyzed in [17], which proposes to use a lace expansion perturbing around non-backtracking random walk. This proof is computer assisted, relying on (1) rigorous numerical upper bounds on various simple random walk integrals as proved by Hara and Slade [25]; and (2) a verification that the numerical conditions in [17] hold true. These two ingredients are implemented in two Mathematica notebooks that can be downloaded from the website of the first author.

The main steps of this paper are (a) to derive a non-backtracking lace expansion for the percolation two-point function; (b) to bound the non-backtracking lace expansion coefficients, thus showing that the general methodology of [17] applies, and (c) to describe the numerical bounds on the coefficients.