Energy of zeros of random sections on Riemann Surface

TL;DR

This paper investigates the asymptotic behavior of the average energy of zeros of random polynomials and holomorphic sections on Riemann surfaces as the degree or number of zeros grows large, comparing to minimal energy configurations.

Contribution

It provides the first detailed asymptotic analysis of the energy of zeros of random holomorphic sections on Riemann surfaces, extending previous results from polynomials to more general line bundles.

Findings

Asymptotic formulas for average energy as N approaches infinity

Comparison of random zeros energy with minimal energy configurations

Extension of results from polynomials to general Riemann surfaces

Abstract

The purpose of this paper is to determine the asymptotic of the average energy of a configuration of N zeros of system of random polynomials of degree N as N tends to infinity and more generally the zeros of random holomorphic sections of a line bundle L over any Riemann surface M. And we compare our results to the well-known minimum of energies.