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The distribution of eigenvalues of GUE and its minors at fixed index

I’ve just uploaded to the arXiv the paper “On the distribution of eigenvalues of GUE and its minors at fixed index”. This is a somewhat technical paper establishing some estimates…

The semicircle law predicts that these gaps have mean close to ; however, due to the aforementioned fluctuations around the classical location, this type of claim is only easy to establish in the “fixed energy”, “averaged energy”, or “averaged index” settings; the “fixed index” case was only achieved by myself as recently as 2013, where I showed that each such gap in fact asymptotically had the expected distribution of the Gaudin law, using manipulations of determinantal processes. For fixed energy, these gaps are in principle well understood, due to previous work of Adler-Nordenstam-van Moerbeke and of Johansson-Nordenstam which show that the spectrum of both matrices is asymptotically controlled by the Boutillier bead process. For low frequencies the triangle inequality argument is unfavorable, and one has to instead use the determinantal kernel of the full minor process, and not just an individual matrix.

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