On the characteristic polynomial of a random unitary matrix

CP Hughes, JP Keating, N O'Connell

Research output: Contribution to journalArticle (Academic Journal)peer-review

87 Citations (Scopus)

Abstract

We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N x N unitary matrix, as N --> infinity. First we show that ln Z / root1/2 ln N, evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for root ln N much less than A greater than or equal to ln N. For higher-order scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A = ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy.
Translated title of the contributionOn the characteristic polynomial of a random unitary matrix
Original languageEnglish
Pages (from-to)429 - 451
Number of pages23
JournalCommunications in Mathematical Physics
Volume220 (2)
DOIs
Publication statusPublished - Jul 2001

Bibliographical note

Publisher: Springer-Verlag
Other identifier: IDS number 455HV

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