Salle 5, Site Marcelin Berthelot
Open to all
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Abstract

We give further proofs of the fact that, for a random hyperbolic surface of large genus, the spectral hole is close to 1/4.

In the first hour, we address the problem of the existence of infinitely many topological types of periodic geodesics in the trace formula. To overcome this, we need to restrict our entire probabilistic model to surfaces that do not contain "tangles". This has the effect of removing a set of surfaces of small probability, and drastically reducing the number of topological types encountered in the trace formula. This step requires the introduction of a "Moebius function" specially designed to eliminate tangles.

In the second hour, we explain how to prove the Friedman-Ramanujan property.

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

N. Anantharaman, L. Monk, A Moebius inversion formula to discard tangled hyperbolic surfaces, https://arxiv.org/abs/2401.01601

Corrected exercises and notes taken during the presentations are available on Thibaut Lemoine's page