Percolation: Where physics and mathematics meet for coffee

We talk to Ralph Kenna and Bertrand Berche about their paper on Universal finite-size scaling for percolation theory in high dimensions published in Journal of Physics A: Mathematical and Theoretical.  They talk about the importance of percolation in mathematics and physics.  And the historical importance of coffee to mathematicians and physicists at the Scottish Café in Lviv.

Who are you?

Ralph Kenna and Bertrand Berche

Ralph Kenna and Bertrand Berche

Ralph Kenna is from the Applied Mathematics Research Centre, Coventry University, Coventry, England.

Bertrand Berche is from the Statistical Physics Group, Université de Lorraine, Nancy, France.
We are both founding members of the L4 Collaboration & Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry, Europe.

What prompted you to pursue this field of research?  

Over the past number of years we have been working on scaling associated with critical phenomena in high-dimensional systems. Although in infinite volume, these are trivially described by mean-field theories above their so-called upper critical dimensions, for finite sizes systems the existing theories (which are over 30 years old) have variously been described as “incomplete”, “disappointing” and “not so good”. We have made significant progress and prompted other groups to also revisit this fundamental pillar of statistical physics.

What is this latest paper all about?

Percolation is a famous problem in mathematics and physics. We are all familiar with the process when making coffee, for example, as water filters through ground beans to deliver the essential brew. The process is also important for draining water through soil, for describing how fashion, opinions, or epidemics spread through a population, or how trendy words become popular in a language. Thus percolation is relevant not only to mathematics and physics, but also in chemistry, geology, network science, sociology, epidemiology, and beyond. The study of percolation started in the 1940s and 1950s and it is estimated that about 80,000 academic papers have since been published on the topic.

A version of the percolation problem may be stated as follows. If we pour fluid on a porous material, will the liquid connect top to bottom? Scientists model this by representing the material as a three-dimensional structure made up of sites which can be wet with a certain probability. If only a few sites are wet, there is not enough water to percolate from top to bottom. If the structure is water-sogged, on the other hand, it is likely that percolation occurs. A certain value of the wetting probability marks the threshold between these extremes. This is called the critical point if the system is infinite. The finite-volume counterpart is the pseudocritical point.

There are different ways to construct finite-size systems. Those with a cubic shape are close to real-life, condensed matter and have open or free boundaries. We can also construct periodic systems which more closely imitate infinitely large systems in the sense that they have no “ends” (just like the surface of a doughnut has no edges). Either set of boundary conditions, free or periodic, should deliver the same results when the system is allowed to grow to infinite size because an infinitely-large system has no boundaries at all.

But how do the clusters appear at the critical and pseudocritical point? How many clusters are of comparable size to the biggest one? And how do they differ from the space they are embedded in? In other words, what is the fractal dimension of the largest percolating clusters?

This problem has long been well understood in the three-dimensional world in which we live (and in which coffee percolators perform their valued tasks). But it has been rather more mysterious in high dimensions. Specifically, in more than six dimensions, physicists long ago claimed that there is a multitude of clusters that are comparable to the biggest, whose fractal dimension is four. This is called the proliferation scenario. Mathematicians agree – but only when the boundary conditions are free. On the other hand, mathematicians have rigorous proofs that there is only one incipient infinite cluster there when periodic boundary conditions are used and it has a fractal dimension larger than four.

The Scottish Cafe, Lviv

The Scottish Café, Lviv

While the problem of making coffee in seven or more dimensions using an infinitely large machine may appear esoteric, it is actually of profound significance and has applications in a wide range of real-life systems. Firstly, what happens in seven dimensions in short-range models of percolation also happens in lower dimensions if the range of interaction is long enough (so-called “spread-out” percolation). This means that it can model long-range contagion, such as spreading of certain illnesses, computer viruses or malware. Secondly, percolation theory ought to manifest a crucial concept called universality. This means that properties of the bulk system hold independent of microscopic details. In particular, the infinite-size system should have no memory of whether it was grown from a finite sized counterpart using free or periodic boundary conditions.

Herein lies the mystery of percolation theory in high dimensions; physicists (strong believers in universality) claimed the largest clusters should proliferate there, while mathematicians (who have rigorous tools at their disposal) were able to prove that this is not so for systems grown out of periodic boundary conditions. The resulting lack of universality would present a crisis for what is called the renormalization group – one of the founding pillars of modern theoretical physics, on which theories from condensed matter to particle physics are built.

Here the puzzle has been resolved. We have shown that universality resides not at the critical point but at the pseudocritical point. There are, almost surely, no clusters of comparable size to the largest, incipient infinite cluster whose fractal dimensionality is bigger than four, irrespective of whether free or periodic boundary conditions are used. In this sense the mathematicians are right.

But the old physics theory was also correct – it holds for free boundary conditions at the critical point where incipient spanning clusters, whose fractal dimension is four, proliferate.

The work has important implications for the manner in which percolation occurs in realistic but spread-out systems: it is not the critical point but the pseudocritical one which manifests the largest percolating clusters and therefore the fastest contagion. The paper should also restore peace and harmony between physicists and mathematicians, as they discuss seven-dimensional structures (undoubtedly over percolated coffee),

What do you plan to do next?

Here we have studied the statistics of clusters in percolation theory. Next we plan to look at geometrical aspects, such as the shapes of critical clusters, how they wrap and fold inside the embedding space.


The photo was taken outside the famous Scottish Café in Lviv, Ukraine, during the Ising Lectures 2017. The café was made famous in the 1930s as the venue where mathematicians from the Lviv School (Banach, Ulam, et al) collaboratively discussed research problems. The large notebook in which they recorded their solved and unsolved problems became known as the Scottish Book. A copy of the Scottish Book exists in the Café and we have entered into it a problem associated with our paper in JPhysA – challenging mathematicians to rigorously verify aspects of our theory.

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Image of the authors and of the Scottish Café used with permission.

Categories: Journal of Physics A: Mathematical and Theoretical

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