L^p estimates for graph eigenfunctions

Do “chaotic” waves spread out randomly, or can they
concentrate near a point?  In the 70s, Berard gave non-trivial bounds
for the sup-norm of a Laplace eigenfunction on a manifold of negative
sectional curvature; though far from the conjectured bounds for
surfaces of negative curvature, and those predicted by the random-wave
model, the bound has not been improved on since.  Recently, Hassel and
Tacy extended Berard’s result to L^p norms, for all p>6.

In this talk we will focus on the analogous problem for large regular
graphs, and show how to get estimates analogous to Berard and
Hassel-Tacy, for all p>2.  We will also discuss how the methods can be
applied to get Hassel-Tacy bounds for joint eigenfunctions on the
sphere. This is joint work with E. Le Masson.