Percolation Theory and Related Topics

California Institute Of Technology, Pasadena CA

Investigators

Abstract

Many large, complex systems arising in mathematics, physics, and elsewhere undergo phase transitions, where varying a parameter (e.g. the temperature or pressure) that describes the system at a small scale by a small amount through some special value causes an abrupt, qualitative change in the behaviour of the system on a large scale. Beyond the familiar examples of water freezing and boiling, phase transitions also occur in many other systems including ferromagnets, superconductors, superfluids, epidemics, and traffic. In each case, understanding when, how, and why the system undergoes a phase transition is of central importance in both theory and practice. Moreover, the basic mathematical principles underlying the occurrence of such phase transitions have much in common across these diverse situations, and the study of phase transitions has come to be recognised as a rich source of deep and beautiful pure mathematics that is of interest beyond and complementary to its practical origins. This project aims to develop our fundamental understanding of phase transitions and critical phenomena (the special properties exhibited by systems at the point of phase transition) through the study of probabilistic models. The project provides research training opportunities for graduate students. The project focuses on various probabilistic lattice models of statistical mechanics, including percolation, random walks, the Ising model, and the uniform spanning tree. In particular, the project aims to understand how the geometry of the underlying graph (e.g. the dimension of the lattice) influences the critical behaviour of the model, with focuses on quantitative aspects of critical phenomena and the comparison between short-range, long-range, and hierarchical models. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.