Universal Finite-Size Scaling for Percolation Theory in High Dimensions

Abstract

We present a unifying, consistent, finite-size-scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions $d_c$. Behaviour at the critical point is non-universal in $d>d_c=6$ dimensions. Proliferation of the largest clusters, with fractal dimension $4$, is associated with the breakdown of hyperscaling there when free boundary conditions are used. But when the boundary conditions are periodic, the maximal clusters have dimension $D=2d/3$, and obey random-graph asymptotics. Universality is instead manifest at the pseudocritical point, where the failure of hyperscaling in its traditional form is universally associated with random-graph-type asymptotics for critical cluster sizes, independent of boundary conditions.