## Columbia Mathematics Department Colloquium

Quantum Ergodicity
on Large Regular Graphs

by

# Nalini Anantharaman

## Université Paris-Sud (Orsay)

Abstract:

Quantum ergodicity'' usually deals with the study of eigenfunctions of
the Laplacian on Riemannian manifolds, in the high-frequency asymptotics.
The rough idea is that, under certain geometric assumptions (like negative
curvature), the eigenfunctions should become spatially uniformly
distributed, in the high-frequency limit. I will review the many
conjectures in the subject, some of which have been turned into theorems
recently. Physicists like Uzy Smilansky or John Keating have suggested
looking for similar questions and results on large (finite) discrete
graphs. Take a large graph $G=(V, E)$ and an eigenfunction $\psi$ of the
discrete Laplacian -- normalized in $L^2(V)$. What can we say about the
probability measure $|\psi(x)|^2$ ($x\in V$)? Is it close to uniform, or
can it, on the contrary, be concentrated in small sets? I will talk about
ongoing work with Etienne Le Masson, in the case of large regular graphs.