GOE statistics on the moduli space of surfaces of large genus

Zeév Rudnick*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


For a compact hyperbolic surface, we define a smooth linear statistic, mimicking the number of Laplace eigenvalues in a short energy window. We study the variance of this statistic, when averaged over the moduli space $\mathcal{M}_{g}$ of all genus g surfaces with respect to the Weil-Petersson measure. We show that in the double limit, first taking the large genus limit and then the short window limit, we recover GOE statistics for the variance. The proof makes essential use of Mirzakhani’s integration formula.

Original languageEnglish
Pages (from-to)1581-1607
Number of pages27
JournalGeometric and Functional Analysis
Issue number6
StatePublished - Dec 2023


  • Gaussian orthogonal ensemble
  • Laplacian
  • Mirzakhani’s integration formula
  • Moduli space
  • Quantum chaos
  • Random matrix theory
  • Riemann surface
  • Selberg trace formula


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