arXiv subjects
Publications and source records attributed to Will Hide.
We study the spectrum of the Laplacian on two models of random hyperbolic 3-orbifolds, related to the Apollonian group and the super Apollonian group. We determine explicit spectral gaps for these random orbifolds. Moreover, we use our model to investigate the bass note spectrum of the set of hyperbolic 3-orbifolds.
We show that there is a constant $c>0$ such that a genus $g$ closed hyperbolic surface, sampled at random from the moduli space $\mathcal{M}_{g}$ with respect to the Weil-Petersson probability measure, has Laplacian spectral gap at least $\frac{1}{4}-O\left(\frac{1}{g^{c}}\right)$ with probability tending to $1$ as $g\to\infty$. This extends and gives a new proof of a recent result of Anantharaman and Monk proved in the series of works [2,3,5,4,6]. Our approach adapts the polynomial method for the strong convergence of random matrices, introduced by Chen, Garza-Vargas, Tropp and van Handel [19], and its generalization to the strong convergence of surface groups by Magee, Puder and van Handel [41], to the Laplacian on Weil-Petersson random hyperbolic surfaces.
In this note we show that the recent work of Magee, Puder and van Handel [MPvH25] can be applied to obtain an optimal spectral gap result with polynomial error rate for uniformly random covers of closed hyperbolic surfaces. Let $X$ be a closed hyperbolic surface. We show there exists $b,c>0$ such that a uniformly random degree-$n$ cover $X_{n}$ of $X$ has no new Laplacian eigenvalues below $\frac{1}{4}-cn^{-b}$ with probability tending to $1$ as $n\to\infty$.
Given a negatively curved compact Riemannian surface $X$, we give an explicit estimate, valid with high probability as the degree goes to infinity, of the first non-trivial eigenvalue of the Laplacian on random Riemannian covers of $X$. The explicit gap is given in terms of the bottom of the spectrum of the universal cover of $X$ and the topological entropy of the geodesic flow on X. This result generalizes in variable curvature a result of Magee-Naud-Puder for hyperbolic surfaces. We then formulate a conjecture on the optimal spectral gap and show that there exists covers with near optimal spectral gaps using a result of Louder-Magee and techniques of strong convergence from random matrix theory.
We prove that the set of limit points of the set of all spectral gaps of closed arithmetic hyperbolic surfaces equals $[0,\frac{1}{4}]$.
We study topological lower bounds on the number of small Laplacian eigenvalues on hyperbolic surfaces. We show that for any $a>0$, there exists $b>0$ such that if $g<an$ and $X$ has genus $g$ and $n$ cusps, then $X$ has at least $b(2g+n-2)$ small eigenvalues. This saturates the sharp upper bound $2g+n-2$ of Otal and Rosas up to a constant multiplicative factor.
We study the geometry and spectral theory of Weil-Petersson random surfaces with genus-$g$ and $n$ cusps in the large-$n$ limit. We show that for a random hyperbolic surface in $\mathcal{M}_{g,n}$ with $n$ large, the number of small Laplacian eigenvalues is linear in $n$ with high probability. By work of Otal and Rosas [41], this result is optimal up to a multiplicative constant. We also study the relative frequency of simple and non-simple closed geodesics, showing that on random surfaces with many cusps, most closed geodesics with lengths up to $\log(n)$ scales are non-simple. Our main technical contribution is a novel large-$n$ asymptotic formula for the Weil-Petersson volume $V_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)$ of the moduli space $\mathcal{M}_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)$ of genus-$g$ hyperbolic surfaces with $k$ geodesic boundary components and $n-k$ cusps with $k$ fixed, building on work of Manin and Zograf [30].
Let $X$ be a finite-area non-compact hyperbolic surface. We study the spectrum of the Laplacian on random covering surfaces of X and on random unitary bundles over X. We show that there is a constant $c > 0$ such that, with probability tending to 1 as $n \to \infty$, a uniformly random degree-$n$ Riemannian covering surface $X_n$ of $X$ has no Laplacian eigenvalues below $\frac{1}{4}-c\frac{(\log\log\log n)^2}{\log \log n}$ other than those of $X$ and with the same multiplicities. We also show that with probability tending to 1 as $n\to \infty$, a random unitary bundle $E_{\phi}$ over $X$ of rank $n$ has no Laplacian eigenvalues below $\frac{1}{4}-c\frac{(\log\log n)^2}{\log n}$.
We study the number of short geodesics and small eigenvalues on Weil-Petersson random genus zero hyperbolic surfaces with $n$ cusps in the regime $n\to\infty$. Inspired by work of Mirzakhani and Petri \cite{Mi.Pe19}, we show that the random multi-set of lengths of closed geodesics converges, after a suitable rescaling, to a Poisson point process with explicit intensity. As a consequence, we show that the Weil-Petersson probability that a hyperbolic punctured sphere with $n$ cusps has at least $k=o(n)$ arbitrarily small eigenvalues tends to $1$ as $n\to\infty$.
We show that for any $\epsilon>0$, $\alpha\in[0,\frac{1}{2})$, as $g\to\infty$ a generic finite-area genus g hyperbolic surface with $n=O\left(g^{\alpha}\right)$ cusps, sampled with probability arising from the Weil-Petersson metric on moduli space, has no non-zero eigenvalue of the Laplacian below $\frac{1}{4}-\left(\frac{2\alpha+1}{4}\right)^{2}-\epsilon$. For $\alpha=0$ this gives a spectral gap of size $\frac{3}{16}-\epsilon$ and for any $\alpha<\frac{1}{2}$ gives a uniform spectral gap of explicit size.
We prove that if $X$ is a finite area non-compact hyperbolic surface, then for any $\epsilon>0$, with probability tending to one as $n\to\infty$, a uniformly random degree $n$ Riemannian cover of $X$ has no eigenvalues of the Laplacian in $[0,\frac{1}{4}-\epsilon)$ other than those of $X$, and with the same multiplicities. As a result, using a compactification procedure due to Buser, Burger, and Dodziuk, we settle in the affirmative the question of whether there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to $\frac{1}{4}$.