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

Ralph Kenna

AMRC, Coventry University
Percolation theory has been the subject of extensive mathematical and simulational studies and is of relevance in a broad range of fields including physics, chemistry, network science, sociology, epidemiology, and geology. It has been reported that 80,000 papers on the topic have appeared in 60 years, including about one per day on the arXiv.
In 1985, Coniglio presented a scaling theory for percolation in high dimensions, suggesting that the proliferation of spanning clusters is associated with the breakdown of hyperscaling in its traditional form there. In the intervening years, mathematicians have verified Coniglio’s theory, but only for systems with free boundary conditions at the infinite-volume percolation threshold or critical point. Numerical results, by contrast, are ambiguous. High- dimensional percolation theory was an active topic in statistical physics up to about 2004 when it was declared that “percolation in high dimensions is not understood.”
In the meantime, the mathematicians have been busy and have made lots of good progress. In 1997, Aizenman established that Coniglio’s predictions hold for bulk boundary conditions and suggested a different scaling behaviour for periodic boundary conditions. Mathematicians have since verified this, but appear to have diverged from the statistical-physics literature and some recent mathematics reviews don’t cite Coniglio’s theory at all.
A number of questions arise for the physicist. Firstly, Coniglio’s theory is built upon renormalization-group concepts such as Kadanoff rescaling, Fisher’s dangerous irrelevant variables and well as Binder’s thermodynamic length. Why do these not deliver the same scaling for different boundary conditions? Why does Aizenman’s picture depend on boundary conditions? What is the explanation of hyperscaling collapse for periodic boundary conditions?
Here, these questions, and more, will be answered. After discussing the history of the problem and the old theories, we show how a recently developed general theory* for scaling in high dimensions recovers Coniglio’s physics and Aizenman’s mathematical predictions in appropriate regimes. This unifies percolation theory and delivers universality and hyperscaling above the upper critical dimension.
  * Ralph Kenna and Bertrand Berche, Universal Finite-Size Scaling for Percolation Theory in High Dimensions and references therein.
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