Equidistribution for Random Polynomials and Systems of Random Holomorphic Sections

Abstract

This article addresses an equidistribution problem concerning the zeros of systems of random holomorphic sections of positive line bundles on compact K\"{a}hler manifolds and random polynomials on $\mathbb{C}^{m}$ in the setting of the weighted pluripotential theory. For random polynomials, we consider non-orthonormal bases and prove an equidistribution result which is more general than the ones acquired before for non-discrete probability measures. More precisely, our result demonstrates that the equidistribution holds true even when the random coefficients in the basis representation are not independent and identically distributed (i.i.d.), and moreover, they are not constrained to any particular probability distribution. For random holomorphic sections, by extending the concept of a sequence of asymptotically Bernstein-Markov measures introduced by Bayraktar, Bloom and Levenberg in their recent paper to the setting of holomorphic line bundles over compact Kahler manifolds, we derive a global equidistribution, variance estimate and expected distribution theorems related to the zeros of systems of random holomorphic sections for large tensor powers of a fixed holomorphic line bundle for any codimension k, generalizing a previous result of Bayraktar in his 2016 paper and giving also a positive answer to a question posed in the same paper, asking whether the equidistribution is true for non-homogeneous manifolds. For both random holomorphic polynomials on $\mathbb{C}^{m}$ and systems of random holomorphic sections, the variance estimation method detailed in another paper of the author with Bojnik is significant.