Random graph asymptotics on high-dimensional tori

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 Zd 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 highdimensional 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 [8, 9], where the V 2/3 scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on Zd. We also strongly rely on mean-field results for percolation on Zd proved in [14, 15, 16, 20].