Nalini Anantharaman

Quantum ergodicity on graphs : from spectral to spatial delocalization

We will be interested in (de)localization phenomena for eigenfunctions of discrete laplacians on graphs.
After reviewing various notions of localization / delocalization, we will more specifically look at the notion of quantum ergodicity and we will prove (under additional assumptions) the following result : if an infinite tree possesses purely absolutely continuous spectrum, and if this tree is ``approximated'' in a certain sense by large finite graphs, then the eigenfunctions of the latter are more or less equidistributed. Note that this is a deterministic result; for certain classes of random graphs, ``quantum unique ergodicity'' has been proven by
Yau, Huang, Bauerschmidt and Knowles.

(Joint work with E. Le Masson, M. Sabri)