The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos - The $L^2$-phase

Abstract

We study the characteristic polynomial of Haar distributed random unitary matrices. We show that after a suitable normalization, as one increases the size of the matrix, powers of the absolute value of the characteristic polynomial as well as powers of the exponential of its argument converge in law to a Gaussian multiplicative chaos measure for small enough real powers. This establishes a connection between random matrix theory and the theory of Gaussian multiplicative chaos.

Some changes to the first version: restriction to real powers and addition of the phase of the characteristic polynomial

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• Mesoscopic linear statistics of Wigner matrices
• On the logarithm of the characteristic polynomial of the Ginibre ensemble