Nearest-neighbor percolation function is continuous for $d>10$

Robert Fitzner, Remco van der Hofstad

Published 2015-06-26Version 1

We study nearest-neighbor percolation in dimensions $d\geq11$, and prove that it displays mean-field behavior. Indeed, we prove that the infrared bound holds, which implies the finiteness of the percolation triangle diagram. The finiteness of the triangle, in turn, implies the existence and mean-field values of various critical exponents, such as $\gamma=1, \beta=1, \delta=2$ and various arm exponents. In particular, our results show that the percolation function is continuous. Such results have been obtained in Hara and Slade (1990,1994) for nearest-neighbor percolation in dimension $d\geq 19$, so that we bring the dimension above which mean-field behaviour is rigorously proved down from $19$ to $11$. Universality arguments predict that the upper critical dimension, above which our results apply, is equal to $d_c=6$. Our results also imply bounds on the critical value of nearest-neighbor percolation on $\mathbb{Z}^d$. We make use of the general method analysed 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. The main steps 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 accompanying paper applies, and (c) to describe the numerical bounds on the coefficients. As a side result, this methodology also allows us to obtain sharp numerical estimates on the percolation critical threshold. 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.