The characteristic polynomial of a random unitary matrix

A probabilistic approach

Paul Bourgade, C. P. Hughes, A. Nikeghbali, M. Yor

Research output: Contribution to journalArticle

Abstract

In this article, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin-Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [8] using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in [8] is now obtained from the classical central limit theorems of probability theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm-type results.

Original languageEnglish (US)
Pages (from-to)45-69
Number of pages25
JournalDuke Mathematical Journal
Volume145
Issue number1
DOIs
StatePublished - Oct 1 2008

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Unitary matrix
Probabilistic Approach
Characteristic polynomial
Random Matrices
Random Polynomials
Sums of Independent Random Variables
Recursion Formula
Mellin Transform
Law of the Iterated Logarithm
Probability Theory
Limit Theorems
Logarithm
Central limit theorem
Argand diagram
Fourier transform
Rate of Convergence
Estimate

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

The characteristic polynomial of a random unitary matrix : A probabilistic approach. / Bourgade, Paul; Hughes, C. P.; Nikeghbali, A.; Yor, M.

In: Duke Mathematical Journal, Vol. 145, No. 1, 01.10.2008, p. 45-69.

Research output: Contribution to journalArticle

Bourgade, Paul ; Hughes, C. P. ; Nikeghbali, A. ; Yor, M. / The characteristic polynomial of a random unitary matrix : A probabilistic approach. In: Duke Mathematical Journal. 2008 ; Vol. 145, No. 1. pp. 45-69.
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