Quantum ergodicity on large graphs
Advances in Homogeneous Dynamics May 11, 2015 - May 15, 2015
Location: MSRI: Simons Auditorium
probabilistic methods in ergodicity
compact Riemannian manifold
quantum variance of operators
negative curvature manifolds
37-XX - Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX]
37D40 - Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
We study eigenfunctions of the discrete laplacian on large regular graphs, and prove a ``quantum ergodicity'' result for these eigenfunctions : for most eigenfunctions $\psi$, the probability measure $|\psi(x)|^2$, defined on the set of vertices, is close to the uniform measure.
Although our proof is specific to regular graphs, we'll discuss possibilities of adaptation to more general models, like the Anderson model on regular graphs.
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