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arXiv · 1602.08875

The maximum of the CUE field

Elliot Paquette·Ofer Zeitouni
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Abstract

Let $U_N$ denote a Haar Unitary matrix of dimension N, and consider the field \[ {\bf U}(z) = \log |\det(1-zU_N)| \] for z in the unit disk. Then, \[ \frac{\max_{|z|=1} {\bf U}(z) -\log N + \frac{3}{4} \log\log N} {\log\log N} \to 0 \] in probability. This provides a verification up to second order of a conjecture of Fyodorov, Hiary and Keating, improving on the recent first order verification of Arguin, Belius and Bourgade.

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Elliot Paquette, Ofer Zeitouni. 2016-02-29. The maximum of the CUE field. https://arxiv.org/abs/1602.08875

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