Abstract
We begin by demonstrating that, for a random hyperbolic surface of large genus, the spectral hole is close to 1/4, with probability tending towards 1. The "trace method" consists in controlling the spectral hole by the number of large periodic geodesics, here using the Selberg trace formula.
It is necessary to get rid of exponential contributions that come from the presence of 0 as a "trivial" eigenvalue on the spectral side, and simple geodesics on the geometric side. We first explain the method of Wu & Xue, who demonstrate that the spectral hole is greater than 3/16. To go further, however, we need to get rid of the exponential contributions that come from all possible types of periodic geodesic topologies: a seemingly insurmountable calculation. We take up Friedman's idea to "get around" this problem: the aim from now on will be to make an asymptotic development to any order of the "volume functions", then to show that the coefficients have the Friedman-Ramanujan property, without explicitly calculating these coefficients. It is this property, together with the choice of a specific test function, that will enable us to cancel out the exponential terms.
References
N. Anantharaman, L. Monk, Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps https://arxiv.org/abs/2304.02678
N. Anantharaman, L. Monk, Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps II, in preparation
N. Anantharaman, L. Monk,Spectral gaps of random hyperbolic surfaces, https://arxiv.org/abs/2403.12576
Y. Wu, Y. Xue, Random hyperbolic surfaces of large genus have first eigenvalues greater than 3/16 - epsilon, GAFA 2022
M. Lipnowski, A. Wright, Towards optimal spectral gaps in large genus, Ann. Probab. 2024
