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Equidistribution for Random Polynomials and Systems of Random Holomorphic Sections

Ozan Günyüz

TL;DR

The paper develops equidistribution results for zeros of random polynomials and random holomorphic sections within weighted pluripotential theory on compact Kähler manifolds. It relaxes common probabilistic assumptions, allowing general bases and non-i.i.d. coefficients, and introduces a framework of asymptotically Bernstein-Markov measures to study high-degree limits. The authors derive explicit formulas for the expected zero currents, establish variance bounds, and prove almost-sure convergence to the Monge–Ampère products of weighted extremal functions, extending known results to non-homogeneous settings. The results unify and broaden previous work (e.g., Bayraktar) by covering codimension-1 to codimension-$k$ zeros and demonstrating global equidistribution for random systems on both $\,\mathbb{C}^m$ and compact Kähler manifolds, with concrete verification for Gaussian, Fubini–Study, and i.i.d. coefficient models.

Abstract

This article addresses an equidistribution problem concerning the zeros of systems of random holomorphic sections of positive line bundles on compact Kähler 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.

Equidistribution for Random Polynomials and Systems of Random Holomorphic Sections

TL;DR

The paper develops equidistribution results for zeros of random polynomials and random holomorphic sections within weighted pluripotential theory on compact Kähler manifolds. It relaxes common probabilistic assumptions, allowing general bases and non-i.i.d. coefficients, and introduces a framework of asymptotically Bernstein-Markov measures to study high-degree limits. The authors derive explicit formulas for the expected zero currents, establish variance bounds, and prove almost-sure convergence to the Monge–Ampère products of weighted extremal functions, extending known results to non-homogeneous settings. The results unify and broaden previous work (e.g., Bayraktar) by covering codimension-1 to codimension- zeros and demonstrating global equidistribution for random systems on both and compact Kähler manifolds, with concrete verification for Gaussian, Fubini–Study, and i.i.d. coefficient models.

Abstract

This article addresses an equidistribution problem concerning the zeros of systems of random holomorphic sections of positive line bundles on compact Kähler manifolds and random polynomials on 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 and systems of random holomorphic sections, the variance estimation method detailed in another paper of the author with Bojnik is significant.
Paper Structure (14 sections, 19 theorems, 103 equations)

This paper contains 14 sections, 19 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Some Auxiliary Deterministic Results
  3. Probabilistic Model
  4. Equidistribution in $1$-codimensional Case
  5. Expected Distribution of Zeros
  6. Variance Estimate
  7. Some Special Distributions
  8. Gaussian and Fubini-Study measures
  9. Random Holomorphic Polynomials with i.i.d. coefficients
  10. Global Equidistribution
  11. Deterministic Setting
  12. Kähler geometric preliminaries
  13. Pluripotential theory on compact Kähler manifolds
  14. Randomization

Key Result

Lemma 2.1

Given a sequence $\{v_{n}\}_{n=1}^{\infty}$ of subharmonic functions on a domain $G\subset \mathbb{R}^{m},\,\,m \geq 2$, which are locally uniformly bounded above on $G$, and a continuous and subharmonic function $v$ on $G$, assume that the following two conditions hold Then $v_{n} \rightarrow v$ in $L^{1}_{loc}(G)$.

Theorems & Definitions (26)

  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • Theorem 4.3
  • proof
  • Corollary 4.4
  • Theorem 4.5
  • ...and 16 more