THE SEARCH FOR UNIVERSALITY IN FINITE-SIZE SCALING OF PERCOLATION THEORY IN HIGH DIMENSIONS

Ralph Kenna (Personal webpage)

Percolation theory has been the subject of extensive mathematical andsimulational studies and is of relevance in a broad range of fields includingphysics, chemistry, network science, sociology, epidemiology, andgeology. It has been reported that 80,000 papers on the topic haveappeared in 60 years, including about one per day on the arXiv.

In 1985, Coniglio presented a scaling theory for percolation in highdimensions, suggesting that the proliferation of spanning clusters isassociated with the breakdown of hyperscaling in its traditionalform there. In the intervening years, mathematicians have verifiedConiglio’s theory, but only for systems with free boundaryconditions at the infinite-volume percolation threshold or criticalpoint. Numerical results, by contrast, are ambiguous. High-dimensional percolation theory was an active topic in statisticalphysics up to about 2004 when it was declared that "percolation inhigh dimensions is not understood".

In the meantime, the mathematicians have been busy and have madelots of good progress. In 1997, Aizenman established thatConiglio's predictions hold for bulk boundary conditions andsuggested a different scaling behaviour for periodic boundaryconditions. Mathematicians have since verified this, but appear tohave diverged from the statistical-physics literature and somerecent mathematics reviews don’t cite Coniglio’s theory at all.

A number of questions arise for the physicist. Firstly, Coniglio'stheory is built upon renormalization-group concepts such as Kadanoffrescaling, Fisher’s dangerous irrelevant variables and well asBinder’s thermodynamic length. Why do these not deliver the samescaling for different boundary conditions? Why does Aizenman’spicture depend on boundary conditions? What is the explanation ofhyperscaling collapse for periodic boundary conditions?

Here, these questions, and more, will be answered. After discussing thehistory of the problem and the old theories, we show how a recently developedgeneral theory [1] for scaling in high dimensions recovers Coniglio’sphysics and Aizenman’s mathematical predictions in appropriateregimes. This unifies percolation theory and delivers universalityand hyperscaling above the upper critical dimension.