Random polynomials and pluripotential-theoretic extremal functions

TL;DR

This paper studies the convergence properties of random polynomials with general coefficient distributions, showing they almost surely approach pluripotential extremal functions and their zero distributions, extending to complex manifolds.

Contribution

It introduces a broad class of random polynomials with general coefficient distributions and proves their almost sure convergence to extremal functions and zero currents, extending classical results.

Findings

Almost sure convergence of scaled log-polynomials to extremal functions

Convergence of zero currents to extremal Monge-Ampère measures

Extension of results to polynomial mappings and line bundles

Abstract

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n whose coefficients with respect to a related orthonormal basis are independent, identically distributed complex-valued random variables having a very general distribution (which includes both normalized complex and real Gaussian distributions) and we prove results on a.s. convergence of a sequence 1/n log |H_n| pointwise and in L^1_{loc}(C^m) to V_{K,Q}. In addition we obtain results on a.s. convergence of a sequence of normalized zero currents dd^c [1/n log |H_n|] to dd^c V_{K,Q} as well as asymptotics of expectations of these currents. All these results extend to random polynomial mappings and to a more general setting of positive holomorphic line bundles over a compact Kahler manifold.