arXiv subjects

Etienne Le Masson

Publications and source records attributed to Etienne Le Masson.

9 recordsLinked to original sources

arXiv2023

Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus

We give a quantitative estimate for the quantum mean absolute deviation on hyperbolic surfaces of finite area in terms of geometric parameters such as the genus, number of cusps and injectivity radius. It implies a delocalisation result of quantum ergodicity type for eigenfunctions of the Laplacian on hyperbolic surfaces of finite area that Benjamini-Schramm converge to the hyperbolic plane. We show that this is generic for Mirzakhani's model of random surfaces chosen uniformly with respect to the Weil-Petersson volume. Depending on the particular sequence of surfaces considered this gives a result of delocalisation of most cusp forms or of Eisenstein series.

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arXiv2022

Eigenfunctions and Random Waves in the Benjamini-Schramm limit

We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory. As a result, we present a mathematically precise formulation of Berry's conjecture for a compact negatively curved manifold and formulate a Berry-type conjecture for sequences of locally symmetric spaces. We prove some weak versions of these conjectures. Using ergodic theory, we also analyze the connections of these conjectures to Quantum Unique Ergodicity.

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arXiv2020

Short geodesic loops and $L^p$ norms of eigenfunctions on large genus random surfaces

We give upper bounds for $L^p$ norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus $g \to +\infty$, we show that random hyperbolic surfaces $X$ with respect to the Weil-Petersson volume have with high probability at most one such loop of length less than $c \log g$ for small enough $c > 0$. This allows us to deduce that the $L^p$ norms of $L^2$ normalised eigenfunctions on $X$ are a $O(1/\sqrt{\log g})$ with high probability in the large genus limit for any $p > 2 + \varepsilon$ for $\varepsilon > 0$ depending on the spectral gap $λ_1(X)$ of $X$, with an implied constant depending on the eigenvalue and the injectivity radius.

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arXiv2019

$L^p$ norms and support of eigenfunctions on graphs

This article is concerned with properties of delocalization for eigenfunctions of Schrödinger operators on large finite graphs. More specifically, we show that the eigenfunctions have a large support and we assess their lp-norms. Our estimates hold for any fixed, possibly irregular graph, in prescribed energy regions, and also for certain sequences of graphs such as N-lifts.

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arXiv2017

$L^p$ Norms of Eigenfunctions on Regular Graphs and on the Sphere

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite collection of algebraic rotations of the $2$-sphere. Under mild conditions, such joint eigenfunctions are shown to satisfy for large $p$ the same bounds as those known for Laplace eigenfunctions on a surface of non-positive curvature.

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arXiv2017

Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces

We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Lindenstrauss and the first-named author on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalised averaging operators over discs, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.

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arXiv2015

Quantum Ergodicity and Averaging Operators on the Sphere

We prove quantum ergodicity for certain orthonormal bases of $L^2(\mathbb{S}^2)$, consisting of joint eigenfunctions of the Laplacian on $\mathbb{S}^2$ and the discrete averaging operator over a finite set of rotations, generating a free group. If in addition the rotations are algebraic we give a quantified version of this result. The methods used also give a new, simplified proof of quantum ergodicity for large regular graphs.

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arXiv2013

Quantum ergodicity on large regular graphs

We propose a version of the Quantum Ergodicity theorem on large regular graphs of fixed valency. This is a property of delocalization of "most" eigenfunctions. We consider expander graphs with few short cycles (for instance random large regular graphs). Our method mimics the proof of Quantum Ergodicity on manifolds: it uses microlocal analysis on regular trees.

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arXiv2013

Pseudo-differential calculus on homogeneous trees

In the objective of studying concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on L^2 and give adjoint and product formulas. Finally we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian.

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