Leírás
Előadó: Abért Miklós
Cím: Benjamini-Schramm convergence and eigenfunctions of Riemannian manifolds
Absztrakt: Let M be a compact manifold with negative curvature. The Quantum Unique Ergdodicity 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 is proved and says that this is true for a density 1 subsequence of eigenfunctions. 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. These are Gaussian random sums of elementary eigenfunctions, the graph analogues of which have been studied by Backhausz, Csoka, Gerencsér, Harangi, Szegedy and Virag.
It turns out that the framework of Benjamini-Schramm convergence allows us to give a mathematically exact formulation of Berry’s conjecture, and establishes a relation to Quantum Unique Ergdodicity. It also proves a weak version of Berry’s conjecture.
This is related to the recent result of Backhausz and Szegedy on almost eigenfunctions of random d-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 does not hold for almost eigenfunctions. One of the exciting questions is what kind of global condition negative curvature enforces in this setting.
A joint work with Nicolas Bergeron and Etienne le Masson.