The PI is investigating questions in several areas of discrete probability that often have surprising interconnections. Most 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. For example, in the random cluster model, there are 4 natural critical values of p for each value of q. The PI is continuing his previous investigations of the relations among these values on planar Cayley graphs of groups. Two other models under investigation concern random spanning forests in graphs. One of these is obtained from limits of minimal spanning trees in finite graphs, while the other is from uniform spanning trees. The former is connected to percolation, a special case of the random cluster model. The latter, connected to random walks and potential theory, is much better understood. The PI is working to bring the state of knowledge of the minimal spanning forest closer to that for the uniform spanning forest. There are also many open questions related to the uniform spanning forest that the PI is investigating. When one views uniform spanning forests as determinantal probability measures, there are a large number of new questions that open up. For example, 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.

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; 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 a decade 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 5 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.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Application #
0231224
Program Officer
Keith Crank
Project Start
Project End
Budget Start
2002-06-15
Budget End
2005-05-31
Support Year
Fiscal Year
2002
Total Cost
$122,836
Indirect Cost
Name
Indiana University
Department
Type
DUNS #
City
Bloomington
State
IN
Country
United States
Zip Code
47401