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Although well described by mean-field theory in the thermodynamic limit,scaling has long been puzzling for finite systems in high dimensions. Thisraised questions about the efficacy of the renormalization group andfoundational concepts such as universality, finite-size scaling andhyperscaling, until recently believed not to be applicable above the uppercritical dimension. Significant theoretical progress has been made resolvingthese issues, and tested in numerous simulational studies of spin models. Thisprogress rests upon superlinearity of correlation length, a notion that for along time encountered resistance but is now broadly accepted. Percolationtheory brings added complications such as proliferation of interpenetratingclusters in apparent conflict with suggestions coming from random-graphasymptotics and a dearth of reliable simulational guidance. Here we report onrecent theoretical progress in percolation theory in the renormalization groupframework in high dimensions that accommodates superlinear correlation andrenders most of the above concepts mutually compatible under different boundaryconditions. Results from numerical simulations for free and periodic boundaryconditions which differentiate between previously competing theories are alsopresented. Although still fragmentary, these Monte Carlo results support thenew framework which restores the renormalization group and foundationalconcepts on which it rests.

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