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

Abstract

We prove that nearest-neighbor percolation in dimensions d ≥ 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 γ = 1, β = 1, δ = 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 Such results had previously been obtained in unpublished work by Hara and Slade for nearest-neighbor percolation in dimension d ≥ 19, so that we bring the dimension above which mean-field behavior is rigorously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 Z d , which are provably at most 1.306% off in d = 11. We make use of the general method analyzed in [16], 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 [24]; and (2) a verification that the numerical conditions in [16] 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 [16] applies, and (c) to describe the numerical bounds on the coefficients.In the appendix of this extended version of the paper, we give additional details about the bounds on the NoBLE coefficients that are not given in the article version.

Series expansion of the percolation threshold on hypercubic lattices

“…We obtained these with our new computations, Theorems 4, 5, and 6, and the identity (9). Here we give b r , the coefficients of p r in the series expansion for d pS defined in (6). The leading term σ r corresponds to the mean-field behavior.…”

Scaling Limit for the Ant in High‐Dimensional Labyrinths

“…The study of critical percolation in high dimensions (currently meaning d 11; see [32,33]) saw significant progress through the use of techniques known as lace expansion (for a recent survey, see [41]). Those techniques allowed a deep understanding of critical clusters in high dimensions (see [36,38]) and in particular opened the door to the proof of the Alexander-Orbach conjecture mentioned above.…”

Sharp threshold phenomena in statistical physics

“…Let us also mention that the second item is often called the mean field lower bound. The lower bound is matched (up to constant) for d ≥ 11 [24], but is expected not to be sharp for small values of d (this fact is known in dimension 2). Theorem 3.6 was first proved by Aizenman, Barsky [1] and Menshikov [34] (these two proofs are presented in [27]).…”