Limit Theorems for Orthogonal Polynomials Related to Circular Ensembles

Joseph Najnudel (), Ashkan Nikeghbali () and Alain Rouault ()
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Joseph Najnudel: Université Paul Sabatier
Ashkan Nikeghbali: Universität Zürich
Alain Rouault: Université Versailles-Saint Quentin, LMV, Bâtiment Fermat

Journal of Theoretical Probability, 2016, vol. 29, issue 4, 1199-1239

Abstract: Abstract For a natural extension of the circular unitary ensemble of order n, we study as $$n\rightarrow \infty $$ n → ∞ the asymptotic behavior of the sequence of monic orthogonal polynomials $$(\varPhi _{k,n}, k=0, \ldots , n)$$ ( Φ k , n , k = 0 , … , n ) with respect to the spectral measure associated with a fixed vector, the last term being the characteristic polynomial. We show that, as $$n\rightarrow \infty $$ n → ∞ , the sequence of processes $$(\log \varPhi _{\lfloor nt\rfloor ,n}(1), t \in [0,1])$$ ( log Φ ⌊ n t ⌋ , n ( 1 ) , t ∈ [ 0 , 1 ] ) converges to a deterministic limit, and we describe the fluctuations and the large deviations.

Keywords: Random matrices; Unitary ensemble; Orthogonal polynomials; Large deviation principle; Invariance principle; 15B52; 42C05; 60F10; 60F17 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10959-015-0632-x

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