Probability on Combinatorial Structures

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0406017Lyons The PI is investigating questions in several areas of probability on various combinatorial structures. Many of these questions are set in a group-invariant context and the goal is to understand how geometric or algebraic properties of the group are reflected in probabilistic properties of the processes. The PI is working to establish basic topological properties of higher-dimensional analogues of random forests and to establish conjectures that arise by analogy to percolation. Phase transitions and entropy of other determinantal dynamical systems are also under investigation. Coupling questions are also at the heart of some stochastic comparison inequalities being studied. Basic questions concern comparison of the behavior of random walks in two random environments. The field of statistical physics is concerned to a great extent with mathematical models of phase transitions (e.g., water to ice). Typically the model of space is a fixed lattice, for example, the square lattice in two dimensions or the cubic lattice in three dimensions. This lattice is infinite and possesses the mathematical properties of what is called a group. The simplest model, known as percolation, originated in the study of fluid flow in the ground and gas flow through a gas mask. One asks how far fluid can flow, in particular, whether it can flow arbitrarily far. This, of course, depends on the density of particles that block the flow; there is a phase transition as the density increases, whereby after a certain point, with probability 1, fluid can no longer flow arbitrarily far. One would like to know where that point is and how the probability changes as this critical point is approached. About 15 years ago, several researchers began investigating lattices that are quite different from the usual Euclidean ones that are most familiar and that most closely correspond to our physical world. These new lattices, called nonamenable, are also usually based on groups. Such investigations began out of the usual scientific and mathematical curiosity that drives fundamental research. Within the last 8 years, this area of research, statistical physics on nonamenable groups, has seen an explosion of interest. This area of research turns out to be quite rich and to contain a large number of important fundamental questions whose answers remain unknown. Already, there have been applications to Euclidean lattices of some of the new ideas that have arisen in response to the need to develop new tools for nonamenable groups.