Heat kernel measures on random surfaces

Semyon Klevtsov, Steve Zelditch

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)135-164
Number of pages30
JournalAdvances in Theoretical and Mathematical Physics
Issue number1
StatePublished - 2016

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)


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