Percolation theory

In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added.

This physical question is modelled mathematically as a three-dimensional network of n × n × n vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability p, or closed with probability 1 – p, and they are assumed to be independent.

This problem, called now bond percolation, was introduced in the mathematics literature by Broadbent & Hammersley (1957),[1] and has been studied intensively by mathematicians and physicists since then.

Since the industrial revolution, the economical importance of this source of energy fostered many scientific studies to understand its composition and optimize its use.

In 1942, Rosalind Franklin, who then recently graduated in chemistry from the university of Cambridge, joined the BCURA.

To measure its 'real' density, one was to sink it in a liquid or a gas whose molecules are small enough to fill its microscopic pores.

While trying to measure the density of coal using several gases (helium, methanol, hexane, benzene), and as she found different values depending on the gas used, Rosalind Franklin showed that the pores of coal are made of microstructures of various lengths that act as a microscopic sieve to discriminate the gases.

One question is to understand how a fluid can diffuse in the coal pores, modeled as a random maze of open or closed tunnels.

[11] This indicates that for a given degree distribution, the clustering leads to a larger percolation threshold, mainly because for a fixed number of links, the clustering structure reinforces the core of the network with the price of diluting the global connections.

For example the distribution of the size of clusters at criticality decays as a power law with the same exponent for all 2d lattices.

However, recently percolation has been performed on a weighted planar stochastic lattice (WPSL) and found that although the dimension of the WPSL coincides with the dimension of the space where it is embedded, its universality class is different from that of all the known planar lattices.

That is, when p < pc, the probability that a specific point (for example, the origin) is contained in an open cluster (meaning a maximal connected set of "open" edges of the graph) of size r decays to zero exponentially in r. This was proved for percolation in three and more dimensions by Menshikov (1986) and independently by Aizenman & Barsky (1987).

The main result for the supercritical phase in three and more dimensions is that, for sufficiently large N, there is almost certainly an infinite open cluster in the two-dimensional slab ℤ2 × [0, N]d − 2.

Thus the subcritical phase may be described as finite open islands in an infinite closed ocean.

Scaling theory predicts the existence of critical exponents, depending on the number d of dimensions, that determine the class of the singularity.

Substantial progress has been made on two-dimensional percolation through the conjecture of Oded Schramm that the scaling limit of a large cluster may be described in terms of a Schramm–Loewner evolution.

This conjecture was proved by Smirnov (2001)[19] in the special case of site percolation on the triangular lattice.

This is a molecular analog to the common board game Jenga, and has relevance to the broader study of virus disassembly.

[21] Percolation theory has been applied to studies of how environment fragmentation impacts animal habitats[24] and models of how the plague bacterium Yersinia pestis spreads.