|Date: Wednesday, March 29, 2006
Title: Point Processes, Repulsion, and Fair Allocation
Abstract: A random collection of points in space is called a point process. The simplest point process is the Poisson process, where the numbers of points in disjoint regions are independent. 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. (For illustration see the movie at http://stat-www.berkeley.edu/~peres/GAF/dynamics/dynamics.html ).
In the second part of the talk 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. The geometry of fair allocations is much richer: 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. In higher dimensions, it appears that gravitational allocation does better than market forces.
(For more details see http://stat-www.berkeley.edu/~peres/stable/stable.html).
Speaker: Yuval Peres
Institution: UC Berkeley-Statistics