Advances in Theoretical and Mathematical Physics

Volume 20 (2016)

Number 1

Heat kernel measures on random surfaces

Pages: 135 – 164

DOI: http://dx.doi.org/10.4310/ATMP.2016.v20.n1.a2

Authors

Semyon Klevtsov (Mathematisches Institut, Universität zu Köln, Germany; and Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, New York, U.S.A.)

Steve Zelditch (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.; and Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, New York, U.S.A.)

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 Kähler 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.

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