Joint Dept. of Mathematics and CMSA Random Matrix & Probability Theory Seminar: Fluctuations in local quantum unique ergodicity for generalized Wigner matrices


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April 16, 2021 2:00 pm - 3:00 pm
via Zoom Video Conferencing

Patrick Lopatto - IAS

In a disordered quantum system, delocalization can be understood in many ways. One of these is quantum unique ergodicity, which was proven in the random matrix context by Bourgade and Yau. It states that for a given eigenvector and set of coordinates J, the mass placed on J by the eigenvector tends to N^{-1}|J|, the mass placed on those coordinates by the uniform distribution. Notably, this convergence holds for any size of J, showing that the eigenvectors distribute evenly on all scales.

I will present a result which establishes that the fluctuations of these averages are Gaussian on scales where |J| is asymptotically less than N, for generalized Wigner matrices with smooth entries. The proof uses new eigenvector observables, which are analyzed dynamically using the eigenvector moment flow and the maximum principle.

This is joint work with Lucas Benigni.