  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.
