# The spectral edge of random band matrices

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Alexander Sodin, Tel Aviv University
We consider random periodic $N\times N$ band matrices of band width $W$. If the band is wide $(W>>N^{5/6})$, the spectral statistics at the edge behave similarly to those of GUE matrices; in particular, the largest eigenvalue converges in distribution to the Tracy—Widom law. Otherwise, a different limit appears. The results are consistent with the Thouless criterion for localization, adapted to the band matrix setting by Fyodorov and Mirlin.