Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Perturbed Toeplitz operators and radial determinantal processes

Torsten Ehrhardt and Brian Rider

Full-text: Open access


We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criterion for a central limit theorem to hold for angular statistics of the points. The proof exploits an exact formula relating the generating function of such statistics to the determinant of a perturbed Toeplitz matrix.


Nous étudions une classe d’ensembles déterminantaux dans le plan complexe invariants par rotation; cette classe comprend les cas des valeurs propres de matrices gaussiennes aléatoires et des zéros de certaines familles de polynomes aléatoires. Le résultat principal est un critère pour l’existence d’un théorème de la limite centrale pour la statistique des angles entre les points. La preuve utilise une formule exacte reliant la fonction génératrice de telles statistiques au déterminant d’une matrice de Toeplitz perturbée.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 4 (2013), 934-960.

First available in Project Euclid: 2 October 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F05: Central limit and other weak theorems 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]

Random matrices Determinantal processes Toeplitz operators Szegö–Widom limit theorem


Ehrhardt, Torsten; Rider, Brian. Perturbed Toeplitz operators and radial determinantal processes. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 4, 934--960. doi:10.1214/12-AIHP501.

Export citation


  • [1] Y. Ameur, H. Hedenmalm and N. Makarov. Random normal matrices and ward identities. Preprint, 2011. Available at arXiv:1109.5941.
  • [2] Z. D. Bai. Circular law. Ann. Probab. 25 (1997) 494–529.
  • [3] E. Basor and T. Ehrhardt. Asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices. Math. Nachr. 228 (2001) 5–45.
  • [4] A. Böttcher and B. Silbermann. Analysis of Toeplitz Operators, 2nd edition. Springer, Berlin, 2006.
  • [5] S.-J. Chen and J. D. Vaaler. The distribution of values of Mahler’s measure. J. Reine Angew. Math. 540 (2001) 1–47.
  • [6] P. Diaconis and S. Evans. Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 (2001) 2615–2633.
  • [7] T. Ehrhardt. A new algebraic approach to the Szegö–Widom limit theorem. Acta Math. Hungar. 99 (2003) 233–261.
  • [8] T. Ehrhardt. A generalization of Pincus’ formula and Toeplitz operator determinants. Arch. Math. (Basel) 80 (2003) 302–309.
  • [9] P. J. Forrester. Fluctuation formula for complex random matrices. J. Phys. A: Math. and General 32 (1999) 159–163.
  • [10] J. Ginibre. Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6 (1965) 440–449.
  • [11] I. Gohberg and M. G. Krein. Introduction to the Theory of Linear Nonselfadjoint Operators on Hilbert Space. Transl. Math. Monographs 18. Amer. Math. Soc., Providence, RI, 1969.
  • [12] J. M. Hammersley. The zeros of a random polynomial. In Proc. of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II 89–111. Univ. California Press, Berkeley and Los Angeles, 1956.
  • [13] J. B. Hough, M. Krishnapur, Y. Peres and B. Virág. Determinantal processes and independence. Probab. Surv. 3 (2006) 206–229.
  • [14] C. P. Hughes, J. P. Keating and N. O’Connell. On the characteristic polynomial of a random unitary matrix. Comm. Math. Phys. 220 (2001) 429–451.
  • [15] K. Johansson. On random matrices from the compact classical groups. Ann. Math. 145 (1997) 519–545.
  • [16] A. Y. Karlovich. Some algebras of functions with Fourier coefficients in weighted Orlicz sequence spaces. In Operator Theoretical Methods and Applications to Math. Physics 287–296. Operator Theory: Advances and Applications 147. Birkhäuser, Basel, 2004.
  • [17] M. Krishnapur. From random matrices to random analytic functions. Ann. Probab. 37 (2009) 314–346.
  • [18] O. Macchi. The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7 (1975) 83–122.
  • [19] Y. Peres and B. Virág. Zeros of the i.i.d. Gaussian power series: A conformally invariant determinantal process. Acta Math. 194 (2005) 1–35.
  • [20] B. Rider. Deviations from the circular law. Probab. Theory Related Fields 130 (2004) 337–367.
  • [21] B. Rider and B. Virág. The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. 2007 (2007) Art. ID rnm006-32.
  • [22] A. Soshnikov. Determinantal random fields. Russian Math. Surveys 55 (2000) 923–975.
  • [23] A. Soshnikov. Gaussian limits for determinantal random point fields. Ann. Probab. 30 (2002) 171–181.
  • [24] H. Widom. Asymptotic behavior of block Toeplitz matrices and determinants. II. Adv. in Math. 21 (1976) 1–29.