# 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.

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