Institute for Advanced Study | Spectral Gap of the Laplacian for Random Hyperbolic Surfaces - Part 2 - Nalini Anantharaman @videosfromIAS | Uploaded 3 weeks ago | Updated 5 hours ago
Special Groups and Dynamics Seminar
Topic: Spectral Gap of the Laplacian for Random Hyperbolic Surfaces - Part 2
Speaker: Nalini Anantharaman
Affiliation: Collège de France
Date: September 18, 2024
Although there are several ways to ''choose a compact hyperbolic surface at random'', putting the Weil-Petersson probability measure on the moduli space of hyperbolic surfaces of a given topology is certainly the most natural. The work of M. Mirzakhani has made possible the study of this probabilistic model, providing exact formulas for certain integrals, as well as their asymptotic behaviour in the limit of large genus.
I will be interested in the spectral gap λ1
of the laplacian for a random compact hyperbolic surface, in the limit of large genus g
: in joint work with Laura Monk, we show that asymptotically almost surely, λ11/4−ϵ
for any ϵ0
.
The proof relies on the trace method. We use asymptotic expansions in powers of g−1
for volume functions giving the distribution of the length spectrum, and prove that the coefficients possess the ``Friedman-Ramanujan property" (a notion introduced by J. Friedman in his proof of the Alon conjecture for random regular graphs).
Special Groups and Dynamics Seminar
Topic: Spectral Gap of the Laplacian for Random Hyperbolic Surfaces - Part 2
Speaker: Nalini Anantharaman
Affiliation: Collège de France
Date: September 18, 2024
Although there are several ways to ''choose a compact hyperbolic surface at random'', putting the Weil-Petersson probability measure on the moduli space of hyperbolic surfaces of a given topology is certainly the most natural. The work of M. Mirzakhani has made possible the study of this probabilistic model, providing exact formulas for certain integrals, as well as their asymptotic behaviour in the limit of large genus.
I will be interested in the spectral gap λ1
of the laplacian for a random compact hyperbolic surface, in the limit of large genus g
: in joint work with Laura Monk, we show that asymptotically almost surely, λ11/4−ϵ
for any ϵ0
.
The proof relies on the trace method. We use asymptotic expansions in powers of g−1
for volume functions giving the distribution of the length spectrum, and prove that the coefficients possess the ``Friedman-Ramanujan property" (a notion introduced by J. Friedman in his proof of the Alon conjecture for random regular graphs).
