The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a modification that allows us to improve its applicability in the nearest-neighbor setting on the Z d -lattice for percolation, lattice trees and lattice animals.The NoBLE gives rise to a recursive formula that we study in this paper at a general level. We state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound. In two related papers, we show that these conditions are satisfied for percolation in d ≥ 11, for lattice trees in d ≥ 16 and for lattice animals in d ≥ 18.

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 Such results had previously been obtained in unpublished work by Hara and Slade for nearest-neighbor percolation in dimension d ≥ 19, so that we bring the dimension above which mean-field behavior is rigorously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.

No abstract