Research

Title

Authors

Pavel Bleher, Bernard Shiffman, Steve Zelditch

Comments

Added results on the decay of connected correlations; corrected typographical errors . To appear in the Proceedings of the 1999 MSRI Workshop on Random Matrices and Their Applications

Abstract

This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebro-geometric generalization studied in our prior work involves random holomorphic sections H⁰(M,L^N) of the powers of any positive line bundle L to M over any complex manifold. Our main interest is in the statistics of zeros of k independent sections (generalized polynomials) of degree N as N approaches infinity. We fix a point P and focus on the ball of radius "1 over square root of N" about P. Under a microscope magnifying the ball by the factor of "square root of N", the statistics of the configurations of simultaneous zeros of random k-tuples of sections tends to a universal limit independent of P,M,L. We review this result and generalize it further to the case of pre-quantum line bundles over almost-complex symplectic manifolds (M,J,omega). Following [SZ2], we replace H⁰(M,L^N) in the complex case with the 'asymptotically holomorphic' sections defined by Boutet de Monvel-Guillemin and (from another point of view) by Donaldson and Auroux. Using a generalization to an m-dimensional setting of the Kac-Rice formula for zero correlations together with the results of [SZ2], we prove that the scaling limits of the correlation functions for zeros of random k-tuples of asymptotically holomorphic sections belong to the same universality class as in the complex case.

Format

Title

Authors

Pavel Bleher, Bernard Shiffman, Steve Zelditch

Comments

3 figures

Abstract

We study the limit as N approaches infinity 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 some explicit formulas for pair correlations. In particular, we provide an alternate derivation of Hannay's limit pair correlation function for SU(2) polynomials, and we show that this correlation function holds for all compact Riemann surfaces.

Format

Title

Authors

Pavel Bleher, Bernard Shiffman, Steve Zelditch

Comments

none

Abstract

This note is concerned with the scaling limit as N approaches infinity of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers L^N of any positive holomorphic line bundle L over a compact Kahler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more generally, of the "correlation forms" are universal, i.e. independent of the bundle L, manifold M or point on M.

Format

Title

Authors

Pavel Bleher, Bernard Shiffman, Steve Zelditch

Comments

13 pages, 1 figure

Abstract

In our previous work [math-ph/9904020], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2. The purpose of this paper is to compute these universal limits in all dimensions and codimensions. First, we use a supersymmetry method to express the n-point correlations as Berezin integrals. Then we use the Wick method to give a closed formula for the limit pair correlation function for the point case in all dimensions.

Format

Title

Authors

Pavel Bleher, Xiaojun Di

Comments

31 pages, 2 figures; a revised version of the J. Stat. Phys. paper

Abstract

We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval $(-1,1)$ are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit translation invariant correlations. Then we calculate the correlations between the straightened zeros of the SO(2) random polynomial.

Format

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