Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I

2023/04/05 by Nalini Anantharaman, Anantharaman, Nalini, Laura Monk +1 · 6 citations
Mathematics · #Mathematical Dynamics and Fractals #Geometry and complex manifolds #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2304.02678

Abstract

In this series of articles, we analyse the level-sets of length functions on the moduli space of compact hyperbolic surfaces of fixed genus. This work ultimately culminates in a proof that typical hyperbolic surfaces have an optimal spectral gap. In this first article, we introduce new volume functions V_g^T(l), counting the expected number of closed geodesics of type T and length l on a random hyperbolic surface of genus g. So far, this function has only been considered in the case where the type is simple, in which case it can be expressed as a combination of Weil-Petersson volumes polynomials, as proven by Mirzakhani. We provide an integral expression for V_g^T(l) for any prescribed type T, which we use to prove that V_g^T(l) admits a full asymptotic expansion in powers of 1/g. We then claim that the coefficients in this expansion, as a function of the length variable l, belong to a newly-introduced class of functions called "Friedman-Ramanujan functions". We relate this claim to the study of the spectral gap of the Laplace-Beltrami operator, and prove it when T fills a surface of Euler characteristic 0 or -1, providing a method to explicitly compute all coefficients in the second-order expansion. We conclude by displaying how the presence of tangles (which is an event of vanishing probability) prevents the sum over all types to satisfy the Friedman-Ramanujan property at the second order.

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