Random band matrices in the localization regime

Abstract: The problem of determining the local eigenvalue statistics (LES) for one-dimensional random band matrices (RBM) will be discussed with an emphasis on the localization regime. RBM are real symmetric $(2N+1)\times(2N+1)$ matrices with nonzero entries in a band of width $2N\alpha+1$ 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 $\frac{1}{2}<α≤1$, 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 α passes through $\frac{1}{2}$. In recent works with B. Brodie and with M. Krishna, we have made progress in proving this conjecture for $0\leq \alpha <\frac{1}{2}$. Some of the results by others for the delocalized state with $\frac{1}{2}<\alpha≤1$ will also be described.

mathematical physicsspectral theory

Audience: researchers in the topic

Munich-Copenhagen-Santiago Mathematical Physics seminar

Series comments: The MAS-MP seminar series has now changed name to MCS-MP on our webpage but not on researchseminars.org

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