Heat kernel measures on random surfaces

TL;DR

This paper explores the use of heat kernel measures on the space of positive definite Hermitian matrices to define probability measures on Bergman metrics of Riemann surfaces, analyzing their fluctuations and connections to random zeros.

Contribution

It introduces a novel family of probability measures on Bergman metrics derived from heat kernel measures, linking metric fluctuations to random zeros of holomorphic sections.

Findings

Fluctuations of the random metric match those of random zeros as k and t tend to infinity.

The heat kernel approach provides explicit formulas for one and two point functions.

In the limit t → ∞, the boundary behavior of zeros influences metric fluctuations.

Abstract

The heat kernel on the symmetric space of positive definite Hermitian matrices is used to endow the spaces of Bergman metrics of degree k on a Riemann surface M with a family of probability measures depending on a choice of the background metric. Under a certain matrix-metric correspondence, each positive definite Hermitian matrix corresponds to a Kahler metric on M. The one and two point functions of the random metric are calculated in a variety of limits as k and t tend to infinity. In the limit when the time t goes to infinity the fluctuations of the random metric around the background metric are the same as the fluctuations of random zeros of holomorphic sections. This is due to the fact that the random zeros form the boundary of the space of Bergman metrics.