Videos

Quantum ergodicity on large graphs

May 11, 2015
Keywords:
  • probabilistic methods in ergodicity
  • geodesic flow
  • compact Riemannian manifold
  • quantum variance of operators
  • negative curvature manifolds
  • graph-theoretic generalization
MSC:
  • 37-xx
  • 37Dxx
  • 37D40
  • 37H99
  • 37Axx
  • 37A99
Abstract
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.
Supplementary Materials