arXiv subjects
Publications and source records attributed to Frédéric Naud.
In this paper, we investigate the singular values of a natural family of transfer operators twisted by large random permutation matrices. In the large N limit, we obtain a Weyl law for its singular values, valid asymptotically almost surely with rapid decay. We also extend the so-called polynomial method to an infinite dimensional setting which implies a "smooth" probabilistic Weyl law for singular values.
In this expository paper, we review the history and the recent breakthroughs in the spectral theory of large volume hyperbolic surfaces. More precisely, we focus mostly on the investigation of the first non-trivial eigenvalue $λ_1$ and its possible behaviour in the large volume regime.
Let $(M,g)$ be some smooth, closed, compact Riemannian manifold and $(M_N\mapsto M)_N$ be an increasing sequence of large degree cyclic covers of $M$ that converges when $N\rightarrow +\infty$, in a suitable sense, to some limit $\mathbb{Z}^p$ cover $M_\infty$ over $M$. Motivated by recent works on zeta determinants on random surfaces and some natural questions in Euclidean quantum field theory, we show the convergence of the sequence $ \frac{\log\det_ζ(Δ_{N})}{\text{Vol}(M_N)} $ when $N\rightarrow +\infty$ where $Δ_N$ is the Laplace-Beltrami operator on $M_N$. We also generalize our results to the case of twisted Laplacians coming from certain flat unitary vector bundles over $M$.
We establish a spectral gap for resonances of the Laplacian of random Schottky surfaces, which is optimal according to a conjecture of Jakobson and Naud.
We investigate the behaviour of the regularized determinant of the Laplace-Beltrami operator on compact hyperbolic surfaces when the genus goes to infinity. We show that for all popular models of random surfaces, with high probability as the genus goes to infinity, the determinant has an exponential growth with a universal exponent. Limit results for some moments of the logarithm of the determinant are then derived.
We consider random n-covers $X_n$ of an arbitrary compact hyperbolic surface $X$. We show that in the large n regime and small window limit, the variance of the smooth spectral statistics of the Laplacian twisted by a unitary abelian character, obey the universal laws of GOE and GUE random matrices, depending on wether the character preserves or breaks the time reversal symmetry. We also prove a generalization for higher dimensional twists valued in compact linear groups. These results confirm a conjecture of Berry and is a discrete analog of a recent work of Rudnick for the Weil-Petersson model of random surfaces.
We consider Laplacians with non unitary twists acting on sections of flat vector bundles over compact hyperbolic surfaces. These non self-adjoint Laplacians have discrete spectrum inside a parabola in the complex plane. For representations of the fundamental group of the base surface which are of Teichmüller type, we investigate the high energy limit and give a precise description of the bulk of the spectrum where Weyl's law is satisfied in terms of critical exponents of the representations which are completely determined by the Manhattan curve associated to the Teichmüller deformation. Our main result provides a counting estimate for the eigenvalues outside the bulk with a polynomial improvement over Weyl's law.
Let $X$ be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature -1. For each $n\in\mathbf{N}$, let $X_{n}$ be a random degree-$n$ cover of $X$ sampled uniformly from all degree-$n$ Riemannian covering spaces of $X$. An eigenvalue of $X$ or $X_{n}$ is an eigenvalue of the associated Laplacian operator $Δ_{X}$ or $Δ_{X_{n}}$. We say that an eigenvalue of $X_n$ is new if it occurs with greater multiplicity than in $X$. We prove that for any $\varepsilon>0$, with probability tending to 1 as $n\to\infty$, there are no new eigenvalues of $X_n$ below $\frac{3}{16}-\varepsilon$. We conjecture that the same result holds with $\frac{3}{16}$ replaced by $\frac{1}{4}$.
Let $X=Λ\backslash\mathbb{H}$ be a Schottky surface, that is, a conformally compact hyperbolic surface of infinite area. Let $δ$ denote the Hausdorff dimension of the limit set of $Λ$. We prove that for any compact subset $\mathcal{K} \subset\{\,s\,:\,\Re(s)>\fracδ{2}\,\}$, if one picks a random degree $n$ cover $X_{n}$ of $X$ uniformly at random, then with probability tending to one as $n\to\infty$, there are no resonances of $X_{n}$ in $\mathcal{K}$ other than those already belonging to $X$ (and with the same multiplicity). This result is conjectured to be the optimal one for bounded frequency resonances and is analogous to both Alon's and Friedman's conjectures for random graphs, which are now theorems due to Friedman and Bordenave-Collins, respectively.
We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=Γ\backslash\mathbb{H}$. Let $δ$ be the Hausdorff dimension of the limit set of $Γ$. We say that a resonance of $X_{n}$ is new if it is not a resonance of $X$, and similarly define new eigenvalues of the Laplacian. We prove that for any $ε>0$ and $H>0$, with probability tending to $1$ as $n\to\infty$, there are no new resonances $s=σ+it$ of $X_{n}$ with $σ\in[\frac{3}{4}δ+ε,δ]$ and $t\in[-H,H]$. This implies in the case of $δ>\frac{1}{2}$ that there is an explicit interval where there are no new eigenvalues of the Laplacian on $X_{n}$. By combining these results with a deterministic `high frequency' resonance-free strip result, we obtain the corollary that there is an $η=η(X)$ such that with probability $\to1$ as $n\to\infty$, there are no new resonances of $X_{n}$ in the region $\{\,s\,:\,\mathrm{Re}(s)>δ-η\,\}$.
It is known that locally constant toral extensions of hyperbolic systems can never mix at an exponential rate. In this note we exhibit some examples of non-abelian locally constant compact extensions of the shift map which are exponentially mixing for Holder-$L^2$ observables. The proof rests on a result of Bourgain-Gamburd and a decoupling argument.
We study circle extensions of analytic Anosov maps on the two torus: these are examples of partially hyperbolic maps for which the qualitative ergodic theory is well understood. In this paper we investigate rates of mixing (for the SRB measure) and prove explicit lower bounds involving the topological pressure of two times the unstable Jacobian. In particular we study the case when the extension function ("roof function") is given by a random trigonometric polynomial.
We prove a local trace formula for Anosov flows. It relates Pollicott--Ruelle resonances to the periods of closed orbits. As an application, we show that the counting function for resonances in a sufficiently wide strip cannot have a sublinear growth. In particular, for any Anosov flow there exist strips with infinitely many resonances.
This papers deals with congruence subgroups of convex cocompact subgroups of PSL2(Z). We examine the behaviour of the resonance spectrum when the congruence parameter q goes to infinity: we show a lower bound for the counting function in discs and an upper bound in vertical strips. These results show drastically different behaviour on both sides of the critical line $\Re(s)=δ/2$.
Let $X$ be a convex co-compact hyperbolic surface and let $δ$ be the Hausdorff dimension of the limit set of the underlying discrete group. We show that the density of the resonances of the Laplacian in strips ${σ\leq \re(s) \leq δ}$ with $|\im(s)| \leq T$ is less than $O(T^{1+δ-ε(σ)})$ with $ε>0$ as long as $σ>δ/2$. This improves the fractal Weyl upper bounds of Zworski and supports numerical results obtained for various models of quantum chaotic scattering.
We consider skew extensions of expanding maps by compact Lie groups. For a class of natural invariant measures, we prove an explicit lower bound on the rate of (exponential) mixing involving topological pressure. Proof uses representation theory and some elliptic pde estimates.
Let $Λ$ be a non-elementary convex co-compact fuchsian group which is a subgroup of an arithmetic fuchsian group. We prove that the Laplace operator of the hyperbolic surface $X=Λ\backslash\H$ has infinitely many resonances in an effective strip depending on the dimension of the limit set $δ$. Applications to lower bounds for the hyperbolic lattice point counting problem are derived.
For convex co-compact hyperbolic quotients $X=Γ\backslash\hh^{n+1}$, we analyze the long-time asymptotic of the solution of the wave equation $u(t)$ with smooth compactly supported initial data $f=(f_0,f_1)$. We show that, if the Hausdorff dimension $δ$ of the limit set is less than $n/2$, then $u(t) = C_δ(f) e^{(δ-\ndemi)t} / Γ(δ-n/2+1) + e^{(δ-\ndemi)t} R(t)$ where $C_δ(f)\in C^\infty(X)$ and $||R(t)||=\mc{O}(t^{-\infty})$. We explain, in terms of conformal theory of the conformal infinity of $X$, the special cases $δ\in n/2-\nn$ where the leading asymptotic term vanishes. In a second part, we show for all $\eps>0$ the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip $\{-nδ-\eps<\Re(\la)<δ\}$. As a byproduct we obtain a lower bound on the remainder $R(t)$ for generic initial data $f$.