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Limit Theorems for Orthogonal Polynomials Related to Circular Ensembles

Author

Listed:
  • Joseph Najnudel

    (Université Paul Sabatier)

  • Ashkan Nikeghbali

    (Universität Zürich)

  • Alain Rouault

    (Université Versailles-Saint Quentin, LMV, Bâtiment Fermat)

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.

Suggested Citation

  • Joseph Najnudel & Ashkan Nikeghbali & Alain Rouault, 2016. "Limit Theorems for Orthogonal Polynomials Related to Circular Ensembles," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1199-1239, December.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:4:d:10.1007_s10959-015-0632-x
    DOI: 10.1007/s10959-015-0632-x
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