Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I
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| Format: | Preprint |
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2023
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| _version_ | 1866912983142105088 |
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| author | Anantharaman, Nalini Monk, Laura |
| author_facet | Anantharaman, Nalini Monk, Laura |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_02678 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I Anantharaman, Nalini Monk, Laura Spectral Theory Geometric Topology 58J50, 32G15 (Primary) 05C80, 11F72 (Secondary) 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. |
| title | Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I |
| topic | Spectral Theory Geometric Topology 58J50, 32G15 (Primary) 05C80, 11F72 (Secondary) |
| url | https://arxiv.org/abs/2304.02678 |