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HD-Combinatorics Special Day: "Quantum ergodicity and spectral theory with a discrete flavour" (organized by Elon Lindenstrauss and Shimon Brooks) | Einstein Institute of Mathematics

HD-Combinatorics Special Day: "Quantum ergodicity and spectral theory with a discrete flavour" (organized by Elon Lindenstrauss and Shimon Brooks)

Date: 
Mon, 25/06/2018
Location: 
Feldman Building, Givat Ram
Title for the day: "Quantum ergodicity and spectral theory with a discrete flavour"

9:00-10:50: Shimon Brooks (Bar Ilan), "Delocalization of Graph Eigenfunctions"
14:00-15:50: Elon Lindenstrauss (HUJI), "Quantum ergodicity on graphs and beyond"

See also the Basic Notions by Elon Lindenstrauss @ Ross 70 (16:30).

Abstract for morning session:
We consider eigenfunctions of the discrete Laplacian on large regular graphs, and show several ways in which these eigenfunctions are “spread out”, and do not concentrate too much in small sets. The first result has a positive-entropy flavor, and states that any subset of the graph supporting a positive proportion of the L^2-mass of an eigenfunction, cannot be small— it’s size must be a power of the size of the graph. The second result gives non-trivial bounds on the L^p norms of an eigenfunction, analogous to the Hassell-Tacy bounds for Laplace eigenfunctions on a manifold of negative curvature. These results are deterministic: they hold for every eigenfucntion and all regular graphs satisfying mild non-degeneracy conditions— essentially that the graphs not have too many short cycles.
The first result is joint with E. Lindenstrauss, the second is joint with E. Le Masson.

Abstract for afternoon session:
The Quantum Ergodicity Theorem of Shnirelman, Colin de Verdier, and Zelditch deals with eigenfunctions of the Laplacian in the semiclassical limit, i.e. eigenvalues tending to infinity. On a d+1 regular graph the Laplacian eigenvalues are bounded by d (and assuming the graph has few short loops as we do most eigenvalues are between -2\sqrt{d} to 2\sqrt{d}.)

Despite this Anantharaman and Le Messon gave a quantum ergodicity theorem for a sequence of expander graphs. I will present another proof of this result by Brooks, Le Messon and myself, as well as analogues for eigenfunctions of averaging operators on spheres.