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 high dimensions, or when d>6 for sufficient 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 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 Aizenman (1997), 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 by Borgs, Chayes, van der Hofstad, Slade and Spencer (2005a, 2005b), where the $V^{2/3}$ scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on Z^d. We also strongly rely on mean-field results for percolation on Z^d proved by Hara (1990, 2005), Hara and Slade (1990) and Hara, van der Hofstad and Slade (2003).