TY - JOUR

T1 - GOE statistics on the moduli space of surfaces of large genus

AU - Rudnick, Zeév

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2023/12

Y1 - 2023/12

N2 - 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.

AB - 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.

KW - Gaussian orthogonal ensemble

KW - Laplacian

KW - Mirzakhani’s integration formula

KW - Moduli space

KW - Quantum chaos

KW - Random matrix theory

KW - Riemann surface

KW - Selberg trace formula

UR - http://www.scopus.com/inward/record.url?scp=85175560757&partnerID=8YFLogxK

U2 - 10.1007/s00039-023-00655-6

DO - 10.1007/s00039-023-00655-6

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AN - SCOPUS:85175560757

SN - 1016-443X

VL - 33

SP - 1581

EP - 1607

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 6

ER -