Analysis, Spectra and Number Theory, a Conference in Honor of Peter Sarnak’s 61st Birthday

Quantum ergodicity usually deals with the equidistribution of eigenfunctions of the Laplacian, in the high frequency limit, on a compact Riemannian manifold. Here I will present an attempt to prove a similar theorem on discrete graphs. We consider a sequence of finite graphs, whose size goes to infinity, and would like to know if the eigenfunctions of the discrete laplacian define a probability close to uniform on the vertices. In joint work with Etienne Le Masson, we proved such a result for regular graphs, assuming they are expanders and have few short cycles. Our proof relied on a "phase space analysis" on the regular tree, which used in a crucial way Fourier analysis on regular trees - and thus cannot be adapted as such to non regular graphs. There is a new proof which does not use Fourier analysis, and thus is more susceptible to be adapted to more general situations - one of the main remaining problems is to find the condition that will replace the "expanding" condition. The same can be asked for a Laplacian + potential on regular graphs.