# Limit theorems for orthogonal polynomials related to circular ensembles

Najnudel, Joseph; Nikeghbali, Ashkan; Rouault, Alain (2016). Limit theorems for orthogonal polynomials related to circular ensembles. Journal of Theoretical Probability, 29(4):1199-1239.

## Abstract

For a natural extension of the circular unitary ensemble of order $\mathit{n}$, we study as $\mathit{n} \to \infty$ the asymptotic behavior of the sequence of monic orthogonal polynomials ($\Phi_{\mathit{k,n}}$, $\mathit{k}$ = 0, … ,$\mathit{n}$) with respect to the spectral measure associated with a fixed vector, the last term being the characteristic polynomial. We show that, $\mathit{n} \to \infty$ , the sequence of processes (log $\Phi_{\llcorner\mathit{nt}\lrcorner, \mathit{n}}$ (1), $\mathit{t} \in$ [0,1]) converges to a deterministic limit, and we describe the fluctuations and the large deviations.

## Abstract

For a natural extension of the circular unitary ensemble of order $\mathit{n}$, we study as $\mathit{n} \to \infty$ the asymptotic behavior of the sequence of monic orthogonal polynomials ($\Phi_{\mathit{k,n}}$, $\mathit{k}$ = 0, … ,$\mathit{n}$) with respect to the spectral measure associated with a fixed vector, the last term being the characteristic polynomial. We show that, $\mathit{n} \to \infty$ , the sequence of processes (log $\Phi_{\llcorner\mathit{nt}\lrcorner, \mathit{n}}$ (1), $\mathit{t} \in$ [0,1]) converges to a deterministic limit, and we describe the fluctuations and the large deviations.

## Statistics

### Citations

1 citation in Web of Science®
1 citation in Scopus®

### Altmetrics

Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2016 19 Jan 2017 07:39 20 Jan 2017 01:00 Springer 0894-9840 https://doi.org/10.1007/s10959-015-0632-x

Preview
Content: Accepted Version
Language: English
Filetype: PDF
Size: 304kB
View at publisher

## Article Networks

### TrendTerms

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.