Date and time: 30 march February 2001, 16:00

Venue: Room 513, School of Mathematical Sciences, Queen Mary, University of London

Speaker: Prof. P. Bleher (Indiana Univ.-Purdue Univ. Indianapolis; currently visiting ENS, Paris)

"Universality and scaling of correlations between zeros on complex manifolds"

Abstract: We study the limit as N goes to infty of the correlations between simultaneous zeros of random sections of the powers L^N of a positive holomorphic line bundle L over a compact complex manifold M, when distances are rescaled so that the average density of zeros is independent of N. We show that the limit correlation is independent of the line bundle and depends only on the dimension of M and the codimension of the zero sets. We also provide explicit formulas for pair correlations. As an example important for physical applications, one may think of a system of i.i.d. k complex random SU(m+1) multivariate polynomials f_j(z_1,...,z_m) of m complex variables, k <= m, of degree N. Then, as the degree N of polynomials goes to infinity, there is a limit of correlations between properly scaled common zeros of f_j(z_1,...,z_m). The limiting two point correlation function demonatrates a quadratic repulsion between zeros when k=m=1 (Hannay), neutrality when k=m=2, and attraction when k=m>2. This is a joint work with Bernie Shiffman and Steve Zelditch (Johns Hopkins University).