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Barry Simon (California Institute of Technology)

OPUC with Random Coefficients

I will discuss orthogonal polynomials on the unit circle with random recursion coefficients. After describing my own work on localization, Ifll discuss the work of Stociu and Killip-Stociu on the statistics of zeros of paraorthogonal polynomials and, in particular, the interesting transition from clock to Poisson behavior.
My work follows the earlier work of Kisselev-Last-Simon and for slow decay shows pure point spectrum, for more rapid decay purely ac spectrum and singular continuous spectrum in between.
Killip-Stociu study CMV matrices (a discrete one-dimensional Dirac-type operator) with random decaying coefficients. Under mild assumptions they identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay they obtain the circular beta ensembles of random matrix theory. The temperature -1 appears as the square of the coupling constant.