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4:00 pm Tuesday, May 3, 2011 Stulken Geometry-Analysis Seminar: Eigenvector Localization for Random Band Matrices with Power Law Band Widthby
Jeffrey Schenker (Michigan State University) in HB 453- Random symmetric matrices with entries that vanish outside a band around the diagonal, but otherwise have independent identically distributed matrix elements, were introduced in the physics literature as an effective model of a ”localization/delocalization” transition seen in disordered materials. In this talk, it will be shown that such matrices satisfy a localization condition which guarantees that eigenvectors have strong overlap with only $W^\mu$ standard basis vectors where W is the band width and $\mu$ is a positive exponent. Thus if $W^\mu << N$, with $N$ the size of the matrix, then a typical eigenvector is essentially supported on a vanishing fraction of standard basis vectors. Some open problems and conjectures will also be discussed.
Submitted by damanik@rice.edu |