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 $\mathbf{C},\mathbf{C}'$ depending only on the dimension and the spread-out parameter such that for any $\lambda \in \mathbb{R}$ if the edge probability is $p_c(\mathbb{Z}^d)+\mathbf{C} \lambda n^{-d/3} + o(n^{-d/3})$, then the joint distribution of the largest clusters normalized by $\mathbf{C}' n^{-2d/3}$ converges as $n\to \infty$ to the ordered lengths of excursions above past minimum of an inhomogeneous Brownian motion started at $0$ with drift $\lambda-t$ at time $t\in[0,\infty)$. This canonical limit was identified by Aldous in the context of critical Erd\H{o}s--R\'enyi graphs.