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Open ended seminar: Quantum ergodicity on large graphs March 31, 2015 (02:00 PM PDT - 04:30 PM PDT)
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Location: MSRI: Baker Board Room
Speaker(s) Nalini Anantharaman (Université de Strasbourg)
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Consider a sequence of finite graphs, of fixed valency, whose size goes to infinity. Assuming their girths go to infinity and that they are expanders, Etienne Le Masson and I proved some version of the "quantum ergodicity" phenomenon : for most of the eigenfunctions $psi_j$ of the laplacian, the probability measure $|psi_j(x)|^2$ on the set of vertices is close to the uniform measure.

I first want to sketch the published proof of this result, based on the "pseudo-differential calculus" on the regular tree, defined by Le Masson. This proof is very specific to regular graphs. 

I then want to present a new, simpler proof, that looks as if it could, in principle, be adapted to non-regular graphs. However this extension to general graphs does not seem obvious and raises several interesting questions in ergodic theory.

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