Related papers: Spectral gap for random Schottky surfaces

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma\backslash \mathbb{H}^2$ be the associated hyperbolic surface. Conditional on the generalized Riemann hypothesis for quadratic $L$-functions, we establish a…

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…

Let $(M,g)$ be a closed Riemannian surface with Anosov geodesic flow. We prove the existence of a spectral gap for Pollicott--Ruelle resonances on random finite coverings of $M$ in the limit of large degree, which is expected to be optimal.…

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…

We prove that the imaginary parts of scattering resonances for negatively curved asymptotically hyperbolic surfaces are uniformly bounded away from zero and provide a resolvent bound in the resulting resonance-free strip. This provides an…

In this paper, we show that a random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than $\left(\frac{1}{4}-\left(\frac{1}{n}\right)^{\frac{1}{221}}\right)$, confirming the nearly optimal spectral gap conjecture…

Let $X=\Lambda\backslash\mathbb{H}$ be a Schottky surface, that is, a conformally compact hyperbolic surface of infinite area. Let $\delta$ denote the Hausdorff dimension of the limit set of $\Lambda$. We prove that for any compact subset…

Let $X$ be a closed, connected, oriented surface of genus $g$, with a hyperbolic metric chosen at random according to the Weil--Petersson measure on the moduli space of Riemannian metrics. Let $\lambda_1=\lambda_1(X)$ bethe first non-zero…

We present a numerical method to calculate resonances of Schottky surfaces based on Selberg theory, transfer operator techniques and Lagrange-Chebyshev approximation. This method is an alternative to the method based on periodic orbit…

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…

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…

A lower bound estimate \lambda_2 - \lambda_1 \ge c \lambda_1^{-d / \alpha} (\diam D)^{-d - \alpha} for the spectral gap of the Dirichlet fractional Laplacian on arbitrary bounded domain D is proved. This follows from a variational formula…

We prove that every family of coverings of any infinite-area, convex cocompact hyperbolic surface has uniform spectral gap, provided that the associated Schreier graphs form a family of two-sided expanders. This extends the results of…

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL} (2,\mathbb{Z})$. We establish a uniform and explicit lower bound of the second eigenvalue of the Laplace-Beltrami operator of congruence coverings of the hyperbolic surface $\Gamma…

The first two eigenvalues of the Robin Laplacian are investigated along with their gap and ratio. Conjectures by various authors for arbitrary domains are supported here by new results for rectangular boxes. Results for rectangular domains…

We show that, with very high probability, the random graph Laplacian has simple spectrum. Our method provides a quantitatively effective estimate of the spectral gaps. Along the way, we establish results on affine no-gaps delocalization,…

Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong…

In this note we provide bounds on the spectral gap for the Dirichlet sub-Laplacians on $H$-type groups. We use probabilistic techniques and in particular small deviations of the corresponding hypoelliptic Brownian motion.

We show that the spectral gap of the first cohomological Laplacian $\Delta_1$ for $\operatorname{Sp}_{2n}(\mathbb{Z})$ follows once a slightly stronger assumption holds for some $\operatorname{Sp}_{2m}(\mathbb{Z})$, where $n\geq m$. As an…

We consider a rigidity problem for the spectral gap of the Laplacian on an $RCD(K,\infty)$-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive $K$. For a weighted Riemannian manifold,…