Typical hyperbolic surfaces have an optimal spectral gap

Abstract: The first non-zero Laplace eigenvalue of a hyperbolic surface, or its spectral gap, measures how well-connected the surface is: surfaces with a large spectral gap are hard to cut in pieces, have a small diameter and fast mixing times. For large hyperbolic surfaces (of large area or large genus g, equivalently), we know that the spectral gap is asymptotically bounded above by 1/4. The aim of this talk is to present an upcoming article, joint with Nalini Anantharaman, where we prove that most hyperbolic surfaces have a near-optimal spectral gap. That is to say, we prove that, for any ε>0, the Weil-Petersson probability for a hyperbolic surface of genus g to have a spectral gap greater than 1/4-ε goes to one as g goes to infinity. This statement is analogous to Alon’s 1986 conjecture for regular graphs, proven by Friedman in 2003. I will present our approach, which shares many similarities with Friedman’s work, and introduce new tools and ideas that we have developed in order to tackle this problem.

mathematical physicsspectral theory

Audience: researchers in the topic

Munich-Copenhagen-Santiago Mathematical Physics seminar

Series comments: The MAS-MP seminar series has now changed name to MCS-MP.

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