## Electronic Journal of Probability

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

Christian Webb

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.ejp/1465067210

Digital Object Identifier
doi:10.1214/EJP.v20-4296

Mathematical Reviews number (MathSciNet)
MR3407221

Zentralblatt MATH identifier
1328.15052

Subjects
Primary: 15A52
Secondary: 60G57: Random measures

Rights

#### Citation

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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