Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces

Abstract

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.

arXiv commentv3: 25 pages, 3 figures, the proof in Section 9 has been corrected, to appear in Duke Math. J

References in corpus (9)

• The ergodic theory of lattice subgroups
• Quantum ergodicity on large regular graphs
• Quantum ergodicity of C* dynamical systems
• Small scale quantum ergodicity in negatively curved manifolds
• Equidistribution of cusp forms in the level aspect
• Cheeger constants and $L^2$-Betti numbers
• $L^p$ norms, nodal sets, and quantum ergodicity
• Pseudo-differential calculus on homogeneous trees
• Quantum ergodicity on large graphs

Cited by in corpus (10)

• Eigenstate Thermalization Hypothesis for Wigner Matrices
• Short geodesic loops and $L^p$ norms of eigenfunctions on large genus random surfaces
• Large covers and sharp resonances of hyperbolic surfaces
• Bounds for twisted symmetric square $L$-functions via half-integral weight periods
• Microlocal lifts and quantum unique ergodicity on $GL(2,{\mathbb{Q}}_p)$
• Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus
• Benjamini-Schramm convergence and spectrum of random hyperbolic surfaces of high genus
• New variants of arithmetic quantum ergodicity
• Quantum Ergodicity for compact quotients of ${\mathrm{SL}}_d(\mathbb{R})/\mathrm{SO}(d)$ in the Benjamini-Schramm limit
• Semiclassical behaviour of quantum eigenstates