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 -