Berry's random-wave conjecture as a local limit
Let $(M, g)$ be a closed Riemannian surface of negative curvature, and let $u_j$ be real eigenfunctions, $-\Delta_g u_j = \lambda_j u_j$, with $\lambda_j \to \infty$ and $\int_M u_j^2 \, dV = \operatorname{Area}(M)$. Sample $x$ uniformly from $M$ and an orthonormal frame $e : \mathbb{R}^2 \to T_x M$ uniformly from the orthogonal group acting on such frames, and define the random smooth function $$F_j(y) = u_j\bigl(\exp_x(e\, y / \sqrt{\lambda_j})\bigr), \qquad y \in \mathbb{R}^2.$$ Let $F$ be the isotropic monochromatic random wave: the centred real Gaussian field on $\mathbb{R}^2$ with covariance $\mathbb{E}[F(y) F(z)] = J_0(|y - z|)$, where $J_0$ is the Bessel function of order zero. Berry suggested that high-frequency eigenfunctions of chaotic systems behave like such random superpositions of plane waves. Abért, Bergeron and Le Masson stated this for an orthonormal basis of eigenfunctions of a compact negatively curved manifold as Benjamini–Schramm convergence to the random wave, and showed that this statement implies quantum unique ergodicity. Ingremeau gave a formulation for chaotic billiards in terms of local weak limits, showed that every sequence of eigenfunctions has local weak limits along subsequences, and proved that certain eigenfunctions of the flat torus have the random wave as their only local weak limit along a density-one subsequence. García-Ruiz proved that the local weak limit and Benjamini–Schramm formulations are equivalent on compact manifolds.
Problem. Do the laws of $F_j$ converge, in the topology of smooth convergence on every compact subset of $\mathbb{R}^2$, to the law of $F$, for every such surface and every such sequence $(u_j)$?
Source: Berry (1977), as a heuristic; made precise for compact negatively curved manifolds by Abért, Bergeron and Le Masson (2018), Conjecture 1, and for chaotic billiards by Ingremeau (2021). Stated for surfaces, with a uniformly random frame, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, PR
- Keywords
- AIM Open Applied Problems, Laplace eigenfunctions, quantum chaos, random waves
References
@article{berry1977,
author = {Berry, M. V.},
title = {Regular and irregular semiclassical wavefunctions},
journal = {J. Phys. A},
volume = {10},
number = {12},
year = {1977},
pages = {2083--2091},
doi = {10.1088/0305-4470/10/12/016},
}
@misc{abert2018,
author = {Ab{\'e}rt, Mikl{\'o}s and Bergeron, Nicolas and Le Masson, Etienne},
title = {Eigenfunctions and random waves in the {B}enjamini--{S}chramm limit},
year = {2018},
eprint = {1810.05601},
archivePrefix = {arXiv},
}
@article{ingremeau2021,
author = {Ingremeau, Maxime},
title = {Local weak limits of {L}aplace eigenfunctions},
journal = {Tunis. J. Math.},
volume = {3},
number = {3},
year = {2021},
pages = {481--515},
doi = {10.2140/tunis.2021.3.481},
eprint = {1712.03431},
archivePrefix = {arXiv},
}
@misc{garciaruiz2023,
author = {Garc{\'\i}a-Ruiz, Alba},
title = {A relation between two different formulations of the {B}erry's conjecture},
year = {2023},
eprint = {2305.14906},
archivePrefix = {arXiv},
}
@misc{aim-berry-random-waves,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Berry's random-wave conjecture as a local limit}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/014-berry-random-waves.md},
note = {Entry 014 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00277, "Berry's random-wave conjecture as a local limit", clopen.solutions.