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

TL;DR

This paper proves that nearest-neighbor percolation in dimensions greater than or equal to 11 exhibits mean-field behavior, using a computer-assisted lace expansion method, and improves the known dimension threshold from 19 to 11.

Contribution

It establishes mean-field behavior for nearest-neighbor percolation in dimensions $d\,\geq\,11$, lowering the previous threshold from $19$ using a novel computer-assisted lace expansion approach.

Findings

Mean-field critical exponents are proven to hold in $d\geq 11$.

Sharp bounds on the critical percolation threshold in $d=11$ are obtained.

The methodology extends the dimension threshold for rigorous proofs of mean-field behavior.

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 Such results had previously been obtained in unpublished work by Hara and Slade for nearest-neighbor percolation in dimension $d\geq 19$, so that we bring the dimension above which mean-field behavior is rigorously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 $1.306\%$ off in $d=11$. We make use of the general method analyzed in the accompanying paper "Generalized approach to the non-backtracking lace expansion" by Fitzner and van der Hofstad, which proposes to use a lace expansion perturbing around non-backtracking random walk. This proof is {\em computer-assisted}, relying on (1) rigorous numerical upper bounds on various simple random walk integrals as proved by Hara and Slade (1992) and (2) a verification that the derived numerical conditions 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 applies, and (c) to describe the numerical bounds on the coefficients. In the appendix of this extended version, we give additional details about the bounds that are not given in the article version.