Super-Brownian limits and the $k$-point function for high-dimensional percolation

Abstract

We prove that there exist positive constants $A$ and $V$ such that the high-dimensional critical percolation $k$-point function is given by \[ T_{p_c}(x_1,x_2,\ldots,x_{k}) \sim V^{k-2} A^{2k-3} \sum_{T\in \mathsf{Tr}(k)} \sum_{\substack{\Phi:V(T)\to \mathbb{Z}^d \\ \Phi(i)=x_i \forall 1\leq i \leq k}} \prod_{\substack{u,v\in V(T)\\u\sim v}}G(\Phi(u),\Phi(v)) \] as $\min_{i\neq j}\|x_i-x_j\|\to \infty$, where $\mathsf{Tr}(k)$ is a set of isomorphism class representatives of trees with $k$ labelled leaves $\{1,\ldots,k\}$ and unlabelled internal vertices all of which have degree $3$ and $G$ is the lattice Green's function. This verifies a conjecture of Aizenman and Newman (1984) subject to the usual perturbative conditions needed for convergence of the lace expansion. It follows from this theorem that the law of the cluster of the origin, considered as the counting measure on its range, converges under rescaling to the canonical measure of the integrated super-Brownian excursion. By computing the asymptotics of various more complicated variations on the $k$-point function, we also prove the stronger result that the cluster converges as an embedded metric-measure space to the continuum random tree equipped with its Brownian embedding into $\mathbb{R}^d$. This convergence holds simultaneously with respect to the chemical distance, pivotal distance, and resistance distance on the cluster, which we prove are asymptotic to constant multiples of each other. This resolves conjectures of Hara and Slade (1998) and van der Hofstad and Slade (2003). As a corollary of our results we prove that there exists a positive constant $C$ such that $\mathbb{P}_{p_c}(0\leftrightarrow \mathbb{Z}^d \setminus [-r,r]^d)\sim C r^{-2}$, answering a question of Heydenreich and van der Hofstad (2017).