The maximum of the CUE field

TL;DR

This paper studies the maximum of a specific logarithmic field derived from Haar unitary matrices on the unit circle, confirming a conjecture about its asymptotic behavior up to second order.

Contribution

It verifies a conjecture on the second-order asymptotics of the maximum of the CUE field, extending previous first-order results.

Findings

The maximum of the CUE field behaves as log N minus (3/4) log log N asymptotically.

The normalized maximum converges to zero in probability.

Provides a second-order verification of a conjecture by Fyodorov, Hiary, and Keating.

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.