4/19/2006

Yuval Peres
University of California, Berkeley

A random collection of points in space is called a "point process". Recently, there has been increasing interest in processes that exhibit "repulsion", such as zeros of random polynomials, noncolliding particles and Eigenvalues of random matrices. I will describe the class of determinantal point processes, which exhibit perfect repulsion, and discuss the dynamical meaning of repulsion, see the movie at http://stat-www.berkeley.edu/~peres/GAF/dynamics/dynamics.html .
In the second part of the talk, based on joint work with C. Hoffman and A. Holroyd, I will discuss the problem of "fair allocation": allocating the same area to every point of an isometry-invariant point process. Given such a point process M in the plane, the Voronoi tesselation assigns a polygon (of different area) to each point of M.  Fair allocations, see   http://stat-www.berkeley.edu/~peres/stable/stable.html have a richer geometry. For any Isometry-invariant point process, we show that there is a unique fair allocation that is "stable" in the sense of the Gale-Shapley stable marriage problem.  It turns out that repelling point processes have allocations that are better localized than the Poisson process.