Critical percolation on the discrete torus in high dimensions

Abstract

We consider percolation on the discrete torus ${\mathbb{Z}}_n^d$ at $p_c({\mathbb{Z}}^d)$, the critical value for percolation on the corresponding infinite lattice ${\mathbb{Z}}^d$, and within the scaling window around it. We assume that $d$ is a large enough constant for the nearest neighbor model, or any fixed $d>6$ for spread-out models. We prove that there exist constants $\mathbf{C},{\mathbf{C}}^{\prime }$ depending only on the dimension and the spread-out parameter such that for any $\hat{I}»\in \mathbb{R}$ if the edge probability is $p_c({\mathbb{Z}}^d)+\mathbf{C}\hat{I}»n^{-d/3}+o(n^{-d/3})$, then the joint distribution of the largest clusters normalized by ${\mathbf{C}}^{\prime }{n}^{-2d/3}$ converges as $n\to \infty$ to the ordered lengths of excursions above past minimum of an inhomogeneous Brownian motion started at $0$ with drift $\hat{I}»-t$ at time $t\in [0,\infty )$. This canonical limit was identified by Aldous in the context of critical Erdős--Rényi graphs.

79 pages, 3 figures