Electronic Journal of Probability

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

Christian Webb

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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.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 104, 21 pp.

Accepted: 6 October 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52
Secondary: 60G57: Random measures

Random Unitary Matrix Characteristic Polynomial Gaussian Multiplicative Chaos

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Webb, Christian. The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos - The $L^2$-phase. Electron. J. Probab. 20 (2015), paper no. 104, 21 pp. doi:10.1214/EJP.v20-4296. https://projecteuclid.org/euclid.ejp/1465067210

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