Convergence of random zeros on complex manifolds

TL;DR

This paper proves that the zeros of random Gaussian polynomial systems on complex manifolds almost surely converge to a predictable equilibrium distribution as the polynomial degree increases.

Contribution

It establishes almost sure convergence of zeros of Gaussian polynomial systems to the equilibrium measure under broad conditions.

Findings

Zeros of polynomial systems converge to equilibrium measure

Convergence holds almost surely as degree increases

Results apply to orthonormalized systems on compact subsets

Abstract

We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros of systems of m polynomials of degree N, orthonormalized on a regular compact subset K of C^m, almost surely converge to the equilibrium measure on K as the degree N goes to infinity.