Quantum Ergodicity for compact quotients of ${\rm SL}_d(\mathbb{R})/{\rm SO}(d)$ in the Benjamini-Schramm limit

在局部对称空间的Benjamini-Schramm极限下的量子厄尔米提性

Quantum ergodicity in the Benjamini--Schramm limit for locally symmetric spaces

研究在局部对称空间序列的Benjamini-Schramm极限下，不变微分算子的本征函数的平均去局域化特性。

Benjamini-Schramm与谱收敛 II. 非齐次情形

Benjamini-Schramm and spectral convergence II. The non-homogeneous case

本文将谱收敛与Benjamini-Schramm收敛的等价性从齐次空间扩展至模等距群紧致的空间，并在一致离散性条件下证明了该等价性。

Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit

We introduce the notion of Benjamini-Schramm convergence for quantum graphs. This notion of convergence, intended to play the role of the already existing notion for discrete graphs, means that the restriction of the quantum graph to a randomly chosen ball has a limiting distribution. We prove that any sequence of quantum graphs with uniformly bounded data has a convergent subsequence in this sense. We then consider the empirical spectral measure of a convergent sequence (with general boundary conditions and edge potentials) and show that it converges to the expected spectral measure of the li…

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.

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, Linde…

Benjamini-Schramm convergence and spectrum of random hyperbolic surfaces of high genus

We study geometric and spectral properties of typical hyperbolic surfaces of high genus, excluding a set of small measure for the Weil-Petersson probability measure. We first prove Benjamini-Schramm convergence to the hyperbolic plane H as the genus g goes to infinity. An estimate for the number of eigenvalues in an interval [a,b] in terms of a, b and g is then proven using the Selberg trace formula. It implies the convergence of spectral measures to the spectral measure of H as g $\rightarrow$+$\infty$, and a uniform Weyl law as b $\rightarrow$+$\infty$. We deduce a bound on the number of sma…