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Probability and Stochastic Processes Seminar

Speaker: Peter Hislop (University of Kentucky, Lexington)

Title: An overview of random band matrices

Abstract: I'll describe the local eigenvalue statistics (LES) problem for random band matrices (RBM). RBM are real symmetric $(2N+1) \times (2N+1)$ matrices with nonzero entries in a band of width $2 N^\alpha +1$ about the diagonal, for $0 \leq \alpha \leq 1$.  The nonzero entries are independent, identically distributed random variables. It is conjectured that as $N \rightarrow \infty$ and for $0 \leq \alpha < \frac{1}{2}$, the LES is a Poisson point process, whereas for $\frac{1}{2} < \alpha \leq 1$, the LES is the same as that for the Gaussian Orthogonal Ensemble. This corresponds to a phase transition from a localized to a delocalized state as $\alpha$ passes through $\frac{1}{2}$. In recent works with B. Brodie and with M. Krishna, we have made progress in proving this conjecture for $0 \leq \alpha < \frac{1}{2}$. Results by others for $\frac{1}{2} < \alpha \leq 1$ will also be described.

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Tuesday, October 12, 2021 at 3:00pm to 4:00pm

Virtual Event
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Lectures & Presentations




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Maximilian Pechmann

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