The fifty-year quest for universality in percolation theory in high dimensions

TL;DR

This paper reviews recent theoretical and numerical advances in high-dimensional percolation theory, resolving longstanding issues about universality, finite-size effects, and the applicability of the renormalization group framework.

Contribution

It introduces a new theoretical framework that incorporates superlinear correlation lengths, reconciling previous conflicting theories and supporting the renormalization group approach in high dimensions.

Findings

Numerical simulations support the new theoretical framework.

Superlinear correlation length is now broadly accepted.

The framework unifies concepts under different boundary conditions.

Abstract

Although well described by mean-field theory in the thermodynamic limit, scaling has long been puzzling for finite systems in high dimensions. This raised questions about the efficacy of the renormalization group and foundational concepts such as universality, finite-size scaling and hyperscaling, until recently believed not to be applicable above the upper critical dimension. Significant theoretical progress has been made resolving these issues, and tested in numerous simulational studies of spin models. This progress rests upon superlinearity of correlation length, a notion that for a long time encountered resistance but is now broadly accepted. Percolation theory brings added complications such as proliferation of interpenetrating clusters in apparent conflict with suggestions coming from random-graph asymptotics and a dearth of reliable simulational guidance. Here we report on recent theoretical progress in percolation theory in the renormalization group framework in high dimensions that accommodates superlinear correlation and renders most of the above concepts mutually compatible under different boundary conditions. Results from numerical simulations for free and periodic boundary conditions which differentiate between previously competing theories are also presented. Although still fragmentary, these Monte Carlo results support the new framework which restores the renormalization group and foundational concepts on which it rests.