Random Graph Asymptotics on High-Dimensional Tori

Abstract

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when d > 6 for sufficiently spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V ^2/3 and below by a small constant times ${V^{2/3}(\log{V})^{-4/3}}$ , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on ${\mathbb{Z}^d}$ under which the lower bound can be improved to small constant times ${V^{2/3}}$ , i.e. we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by [1], apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation.

Our method is crucially based on the results in [11, 12], where the ${V^{2/3}}$ scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on ${\mathbb{Z}^d}$ . We also strongly rely on mean-field results for percolation on ${\mathbb{Z}^d}$ proved in [17–20].