Characteristic polynomials for 1D band matrices from the localization side

Characteristic polynomials for 1D band matrices from the localization side

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Tatyana Shcherbina , IAS
Fine Hall 601

The physical conjecture about the crossover for $N\times N$ 1D random band matrices with the band width $W$ states that we get the same behavior of eigenvalues correlation functions as for GUE for $W\gg \sqrt{N}$ (which corresponds to delocalized states), and we get another behavior, which is determined by the Poisson statistics, for $W\ll \sqrt{n}$ (and corresponds to localized states). The question is still open (there are some partial results only), however, the first part of the conjecture was proved for more accessible objects than eigenvalues correlation functions, namely, for the correlation functions of characteristic polynomials. In this talk we complement this result and prove that for $W\ll \sqrt{n}$ the behavior of the second correlation function of characteristic polynomials is different from those for $GUE$.  Joint work with Mariya Shcherbina.