The effect of geometry on phase transitions

Seminar in Mathematics

Abstract: Large systems of interacting components often undergo phase transitions. The most recognisable examples come from physics: a small change in temperature causes ice to melt and magnets to lose their magnetism. Percolation is the mathematician’s caricature of such a system. The components in this model are traditionally arranged as the vertices of a Euclidean lattice. However, in recent decades, mathematicians have been investigating what happens if the components are arranged in some other homogeneous way, say as the vertices of a Cayley graph of another infinite group. For example, how does this arrangement determine the "critical point" where a phase transition occurs? I will describe some recent progress in this area and outline an ongoing project to unify this theory with the celebrated Erdős–Rényi phase transition from combinatorics.