Work

Almost Sure GOE Fluctuations of Energy Levels for Hyperbolic Surfaces of High Genus

Year: 2025

Type: article

Abstract: We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian orthogonal ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. In this note, we show that in this same limit, the (smooth) energy variance for a typical surface is close to the GOE result, a feature called "ergodicity" in the random matrix theory literature.

Source: Annales Henri Poincaré

DOI: 10.1007/s00023-025-01552-4

Authors: Zeév Rudnick, Igor Wigman

Institutions: Tel Aviv University, King's College London

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: European Commission, National Natural Science Foundation of China, Israel Science Foundation, FP7 Ideas: European Research Council