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.

Revised version, 26 pages, no figures

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• Equivalent-neighbor percolation models in two dimensions: crossover between mean-field and short-range behavior
• Finite-size scaling of the random-field Ising model above the upper critical dimension
• Effective-Dimension Theory of Critical Phenomena above Upper Critical Dimensions
• The fifty-year quest for universality in percolation theory in high dimensions
• When correlations exceed system size: finite-size scaling in free boundary conditions above the upper critical dimension
• Universality of percolation at dynamic pseudocritical point
• Phase transitions above the upper critical dimension
• On a previously unpublished work with Ralph Kenna