  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 Such results had previously been obtained in unpublished work by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.
