Gaussian value distribution of chaotic eigenfunctions
In plain words
Berry proposed that a high-frequency standing wave of a chaotic system looks locally like a random superposition of plane waves, so its values follow a bell curve. Even a much weaker consequence, that no eigenfunction has unusually high peaks, is unproven.
Precise statement
Compact negatively curved surface $M$ (or an ergodic billiard), $-\operatorname{Lap} \phi_{j} = \lambda_{j} \phi_{j}$, $||\phi_{j}||_{2} = 1$. Prove that for a density-one subsequence the distribution of $\phi_{j}(x)$, $x$ uniform on $M$, converges to the Gaussian of variance $1/\operatorname{vol}(M)$; a weaker target is the sup-norm bound ||phi_j||_inf <= C_eps lambda_j^eps. Answer: a proof.
What would settle it
A proof of Gaussian moments (e.g. ||phi_j||_4^4 -> 3/vol(M)) for a density-one subsequence on one negatively curved surface.
Status in the literature
The general sup-norm bound is $\lambda^{1/4}$ up to logarithmic improvement; on arithmetic surfaces Iwaniec and Sarnak (1995) proved $\lambda^{5/24+\varepsilon}$.