## Columbia Mathematics Department Colloquium

Quantum Ergodicity

on Large Regular Graphs

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.

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.