Quantum ergodic sequences and equilibrium measures

TL;DR

This paper surveys recent advances in the study of Bergman kernels, equilibrium measures, and zeros of random holomorphic sections, introducing a new definition of quantum ergodic sections and proving their asymptotic properties.

Contribution

It introduces a new definition of quantum ergodic sections for Hermitian line bundles and proves their asymptotic zero distribution, extending prior results to more general settings.

Findings

Asymptotic equilibrium distribution of zeros of quantum ergodic sections

Random sequences and orthonormal bases of sections are quantum ergodic

New definition of quantum ergodic sections for general smooth metrics

Abstract

This is partly a survey article for Constructive Approximation's Special Issue on Approximation and Statistical Physics. It reviews results of B. Shiffman-S.Zelditch, R. Berman. S. Boucksom, D. Witt-Nystrom, T. Bloom, O. Zeitouni and others on Bergman kernels and the asymptotic equilibrium distribution of zeros of random polynomials and more general holomorphic sections of ample line bundles. The article also gives a new definition of `quantum ergodic section of a line bundle' for Hermitian line bundles with general smooth metrics and Bernstein-Markov measurs. It proves the asymptotic equilibrium distribution of zeros of these generalizaed QE sections. It also proves that random sequences and random orthonormal bases of sections are QE.