Universal Finite-Size Scaling for Percolation Theory in High Dimensions

Ralph Kenna, Bertrand Berche

Published 2016-06-01Version 1

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. Behaviour at the critical point is non-universal; cluster proliferation is responsible for the breakdown of hyperscaling there when free boundary conditions are used but not when the boundary conditions are periodic. Universality is instead manifest at the pseudocritical point, where the non-proliferation scenario is independent of boundary conditions. The failure of hyperscaling in its traditional form there is universally ascribed to the growth in sizes of clusters rather than the growth of their number.