Special Seminar

Let M be a compact manifold with negative curvature. The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on M get equidistributed as the eigenvalue tends to infinity. A weaker version, called Quantum Ergodicity by Shnirelman, Colin de Verdière, and Zelditch says that this is true for a density 1 subsequence. On the other hand, a conjecture of Berry, that has not been formulated in a mathematically precise way, says that high eigenvalue eigenfunctions behave like Gaussian random Euclidean waves. Using the framework of Benjamini-Schramm convergence, we give a mathematically exact formulation of Berry’s conjecture. This allows us to establish a relation to Quantum Unique Ergodicity and to prove a version of Berry’s conjecture in a quite specific setting.

Our approach is related to the recent result of Backhausz and Szegedy on almost eigenfunctions of random regular graphs. A major difference is that instead of expander graphs, here one deals with a hyperfinite sequence of manifolds. In particular, opposed to the Backhausz-Szegedy theorem, Berry's conjecture will not hold for almost eigenfunctions.