Description

Percolation theory dates back to the 1950s and addresses the problem of flow of gases and liquids through irregular porous solids e.g. underground oil and gas reservoirs, electron flow through complex media, flow of chemical solutions through sheets (liquid chromatography). It also has application to crack propagation in regular solids and failure modes of complex systems. It is a source of the finest mathematical problems: easily stated but whose solutions are apparently difficult and require new methods. At the same time, many of the problems are of interest to or were originally proposed by scientists rather than mathematicians.

The simplest model is the square lattice in the plane which has sites at points in Z² connected by edges in the axial directions which are open with probability p but otherwise closed, independently for each edge. Flow can take place only along the open edges and we want to know which sites can be reached by fluid entering the lattice at the origin. The characteristic result for this model is that there is a critical threshold p_c such that for p < p_c only a finite number of sites can be reached but for p > p_c an infinite number of sites can be reached. This threshold phenomenon is also found in more complex models and in higher dimensions.

Percolation theory is now an established mathematical discipline. There are major questions which remain largely unanswered but it lies at the heart of the theory of disordered physical systems. After studying the standard models and some of the primary results there are various directions in which a project on this topic can develop.

Prerequisites

Resources

• Percolation by Bela Bollobas and Oliver Riordan,
• Percolation Theory by Geoffrey Grimmett,
• Percolation Theory for Mathematicians by Harry Kesten.

The first chapter of Grimmett's book is available on-line as a pdf file.

email: M. Menshikov