Extensions of Percolation

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Dr Holroyd is working on several natural extensions of the percolation model in mathematical probability. These are: entanglement, rigidity, random surfaces, and percolation on transitive graphs. In the areas of entanglement and rigidity, Holroyd's previous work provided the first rigorous mathematical treatment of the subjects, and Holroyd is now pursuing some of the remaining unsolved problems. For entanglement, Holroyd hopes to answer questions about the sizes of finite entangled graphs, the effect of boundary conditions, and the value of the critical probability. For rigidity, Holroyd is investigating the effect of boundary conditions, and the continuity of the percolation probability. Also, Holroyd is investigating the appearance of random surfaces in the so-called plaquette percolation model (which has mathematical connections with entanglement). Holroyd hopes to answer questions about the values of the critical probabilities for the appearance of infinite surfaces, and about the geometry of such surfaces. Finally, Holroyd is studying the percolation model on general graphs, such as transitive graphs. There has been much recent progress in this field, but many fundamental questions remain unanswered. Holroyd is focusing on proving strict inequalities between critical probabilities for different graphs, and between the critical points for uniqueness and for percolation. Percolation is a branch of probability theory concerned with the large-scale properties of infinite collections of independent random objects. The original motivation for the subject was the study of fluid flow in porous materials, but the mathematical models which arise have many other applications, for example in material science, statistical physics, biology and epidemiology. The models are relatively simple to describe, yet lead to challenging mathematical problems. Percolation is a well-developed and active area of mathematical research, boasting hundreds of papers and several books, yet several of the most fundamental questions remain unanswered. Holroyd's work focuses on several relatively new extensions of percolation. Two of these, entanglement and rigidity, have particularly close connections with physical applications in the science of materials, and have been studied previously by physicists. Some of Holroyd's earlier work provided the first mathematical treatment of these two subjects, and Holroyd is now pursuing some of the remaining unsolved problems in these and other areas.