Average characteristic polynomials of determinantal point processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 283-303.

On s’intéresse au polynôme caractéristique moyen 𝔼[ i=1 N (z-x i )] associé à des variables aléatoires réelles x 1 ,...,x N qui forment un Ensemble Biorthogonal, c’est-à-dire un processus ponctuel déterminantal associé à un opérateur de projection borné et de rang fini. Pour une sous-classe d’Ensembles Biorthogonaux, qui contient les Ensembles Polynômes Orthogonaux et les Ensembles Polynômes Orthogonaux Multiples (de type mixte), nous obtenons une condition suffisante pour que, presque sûrement, la distribution limite de ses zéros coincide avec la distribution limite des variables aléatoires, quand N tend vers l’infini. De plus, cette condition s’avère être également suffisante pour améliorer la convergence en moyenne en convergence presque sûre pour les moments de la mesure empirique associée au processus ponctuel déterminantal. En application, on obtient avec des théorèmes de Voiculescu une description pour les distributions limites des zéros des polynômes d’Hermite et de Laguerre multiples, en termes de convolutions libres de lois classiques avec des mesures atomiques, ainsi que des équations algébriques explicites pour leurs transformées de Cauchy–Stieltjes.

We investigate the average characteristic polynomial 𝔼[ i=1 N (z-x i )] where the x i ’s are real random variables drawn from a Biorthogonal Ensemble, i.e. a determinantal point process associated with a bounded finite-rank projection operator. For a subclass of Biorthogonal Ensembles, which contains Orthogonal Polynomial Ensembles and (mixed-type) Multiple Orthogonal Polynomial Ensembles, we provide a sufficient condition for its limiting zero distribution to match with the limiting distribution of the random variables, almost surely, as N goes to infinity. Moreover, such a condition turns out to be sufficient to strengthen the mean convergence to the almost sure one for the moments of the empirical measure associated to the determinantal point process, a fact of independent interest. As an application, we obtain from Voiculescu’s theorems the limiting zero distribution for multiple Hermite and multiple Laguerre polynomials, expressed in terms of free convolutions of classical distributions with atomic measures, and then derive explicit algebraic equations for their Cauchy–Stieltjes transform.

DOI : 10.1214/13-AIHP572
Classification : 60B10, 60F15, 60B20, 60K35, 26C10
Mots clés : determinantal point processes, average characteristic polynomials, strong law of large numbers, random matrices, multiple orthogonal polynomials
@article{AIHPB_2015__51_1_283_0,
     author = {Hardy, Adrien},
     title = {Average characteristic polynomials of determinantal point processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {283--303},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {1},
     year = {2015},
     doi = {10.1214/13-AIHP572},
     mrnumber = {3300971},
     zbl = {06412905},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/13-AIHP572/}
}
TY  - JOUR
AU  - Hardy, Adrien
TI  - Average characteristic polynomials of determinantal point processes
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2015
SP  - 283
EP  - 303
VL  - 51
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/13-AIHP572/
DO  - 10.1214/13-AIHP572
LA  - en
ID  - AIHPB_2015__51_1_283_0
ER  - 
%0 Journal Article
%A Hardy, Adrien
%T Average characteristic polynomials of determinantal point processes
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2015
%P 283-303
%V 51
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/13-AIHP572/
%R 10.1214/13-AIHP572
%G en
%F AIHPB_2015__51_1_283_0
Hardy, Adrien. Average characteristic polynomials of determinantal point processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 283-303. doi : 10.1214/13-AIHP572. http://archive.numdam.org/articles/10.1214/13-AIHP572/

[1] G. W. Anderson, A. Guionnet and O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge, 2010. | MR | Zbl

[2] G. W. Anderson and O. Zeitouni. A law of large numbers for finite-range dependent random matrices. Comm. Pure Appl. Math. 61 (2008) 1118–1154. | DOI | MR | Zbl

[3] J. Arvesú, J. Coussement and W. Van Assche. Some discrete multiple orthogonal polynomials. J. Comput. Appl. Math. 153 (2003) 19–45. | DOI | MR | Zbl

[4] A. I. Aptekarev, A. Branquinho and W. Van Assche. Multiple orthogonal polynomials for classical weights. Trans. Amer. Math. Soc. 355 (2003) 3887–3914. | DOI | MR | Zbl

[5] M. Bender, S. Delvaux and A. B. J. Kuijlaars. Multiple Meixner–Pollaczek polynomials and the six-vertex model. J. Approx. Theory 163 (2011) 1606–1637. | DOI | MR | Zbl

[6] H. Bercovici and D. Voiculescu. Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42 (1993) 733–773. | DOI | MR | Zbl

[7] P. Biane. On the free convolution with a semi-circular distribution. Indiana Univ. Math. J. 46 (1997) 705–718. | DOI | MR | Zbl

[8] P. M. Bleher, S. Delvaux and A. B. J. Kuijlaars. Random matrix model with external source and a constrained vector equilibrium problem. Comm. Pure Appl. Math. 64 (1) (2011) 116–160. | MR | Zbl

[9] P. M. Bleher and A. B. J. Kuijlaars. Random matrices with external source and multiple orthogonal polynomials. Int. Math. Res. Not. 3 (2004) 109–129. | DOI | MR | Zbl

[10] P. M. Bleher and A. B. J. Kuijlaars. Integral representations for multiple Hermite and multiple Laguerre polynomials. Ann. Inst. Fourier (Grenoble) 55 (2005) 2001–2014. | Numdam | MR | Zbl

[11] A. Borodin. Biorthogonal ensembles. Nuclear Phys. B 536 (3) (1999) 704–732. | MR | Zbl

[12] J. Breuer and M. Duits. The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles. Manuscript, 2013. Available at arXiv:1301.2061. | MR | Zbl

[13] E. Coussement and W. Van Assche. Some classical multiple orthogonal polynomials. J. Comput. Appl. Math. 127 (2001) 317–347. | DOI | MR | Zbl

[14] E. Daems and A. B. J. Kuijlaars. Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions. J. Approx. Theory 146 (2007) 91–114. | DOI | MR | Zbl

[15] P. Deift. Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics 3. Amer. Math. Soc., Providence, RI, 1999. | MR | Zbl

[16] S. Delvaux. Average characteristic polynomials for multiple orthogonal polynomial ensembles. J. Approx. Theory 162 (2010) 1033–1067. | DOI | MR | Zbl

[17] P. Desrosiers and P. J. Forrester. A note on biorthogonal ensembles. J. Approx. Theory 152 (2) (2008) 167–187. | MR | Zbl

[18] M. Duits, D. Geudens and A. B. J. Kuijlaars. A vector equilibrium problem for the two-matrix model in the quartic/quadratic case. Nonlinearity 24 (2011) 951–993. | DOI | MR | Zbl

[19] M. Duits and A. B. J. Kuijlaars. Universality in the two matrix model: A Riemann–Hilbert steepest descent analysis. Comm. Pure Appl. Math. 62 (2009) 1076–1153. | DOI | MR | Zbl

[20] M. Duits, A. B. J. Kuijlaars and M. Y. Mo. The Hermitian two-matrix model with an even quartic potential. Memoirs Amer. Math. Soc. 217 (1022) (2012) 1–105. | MR | Zbl

[21] M. Duits, A. B. J. Kuijlaars and M. Y. Mo. Asymptotic analysis of the two matrix model with a quartic potential. In Proceedings of the MSRI Semester “Random Matrix Theory, Interacting Particle Systems and Integrable Systems.” To appear. Available at arXiv:1210.0097. | MR | Zbl

[22] A. Edelman and N. R. Rao. The polynomial method for random matrices. Found. Comput. Math. 8 (2008) 649–702. | DOI | MR | Zbl

[23] W. Van Assche, J. S. Geronimo and A. B. J. Kuijlaars. Riemann–Hilbert problems for multiple orthogonal polynomials. In NATO ASI Special Functions 2000. Current Perspective and Future Directions 23–59. J. Bustoz, M. E. H. Ismail and S. K. Suslov (Eds). Nato Science Series II 30. Kluwer Academic Publishers, Dordrecht, 2001. | MR | Zbl

[24] I. Gohberg, S. Goldberg and N. Krupnik. Traces and Determinants of Linear Operators. Birkhäuser, Basel, 2000. | DOI | MR | Zbl

[25] A. Hardy and A. B. J. Kuijlaars. Large deviations for a non-centered Wishart matrix. Random Matrices Theory Appl. 2 (1) (2013) 1250016. | DOI | MR | Zbl

[26] J. B. Hough, M. Krishnapur, Y. Peres and B. Virág. Determinantal processes and independence. Probab. Surveys 3 (2006) 206–229. | DOI | MR | Zbl

[27] M. E. H. Ismail. Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications 98. Cambridge Univ. Press, Cambridge, 2005. | MR | Zbl

[28] K. Johansson. Random matrices and determinantal processes. In Mathematical Statistical Physics: Lecture Notes of the Les Houches Summer School 2005 1–55. Bovier et al. (Eds.). Elsevier, Amsterdam, 2006. | MR | Zbl

[29] K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000) 437–476. | DOI | MR | Zbl

[30] K. Johansson. Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. 153 (2001) 259–296. | DOI | MR | Zbl

[31] W. König. Orthogonal polynomial ensembles in probability theory. Probab. Surveys 2 (2005) 385–447. | DOI | MR | Zbl

[32] A. B. J. Kuijlaars. Multiple orthogonal polynomial ensembles. In Recent Trends in Orthogonal Polynomials and Approximation Theory 155–176. Contemp. Math. 507. Amer. Math. Soc., Providence, RI, 2010. | MR | Zbl

[33] A. B. J. Kuijlaars. Multiple orthogonal polynomials in random matrix theory. In Proceedings of the International Congress of Mathematicians III 1417–1432. Hindustan Book Agency, New Delhi, 2010. | MR | Zbl

[34] A. B. J. Kuijlaars and K. T.-R. Mclaughlin. A Riemann–Hilbert problem for biorthogonal polynomials. J. Comput. Appl. Math. 178 (2005) 313–320. | DOI | MR | Zbl

[35] A. B. J. Kuijlaars, A. Martínez-Finkelshtein and F. Wielonsky. Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights. Comm. Math. Phys. 286 (2009) 217–275. | DOI | MR | Zbl

[36] A. B. J. Kuijlaars and P. Román. Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths. J. Approx. Theory 162 (2010) 2048–2077. | DOI | MR | Zbl

[37] V. Lysov and F. Wielonsky. Strong asymptotics for multiple Laguerre polynomials. Constr. Approx. 28 (2008) 61–111. | DOI | MR | Zbl

[38] K. A. Muttalib. Random matrix models with additional interactions. J. Phys. A 28 (1995) L159–164. | MR

[39] L. Pastur and M. Shcherbina. Eigenvalue Distribution of Large Random Matrices. Mathematical Surveys and Monographs 171. Amer. Math. Soc., Providence, RI, 2011. | DOI | MR | Zbl

[40] P. Nevai. Orthogonal polynomials. Mem. Amer. Math. Soc. 213 (1979). | MR | Zbl

[41] B. Simon. Weak convergence of CD kernels and applications. Duke Math. J. 146 (2009) 305–330. | DOI | MR | Zbl

[42] A. Soshnikov. Determinantal random point fields. Russian Math. Surveys 55 (2000) 923–975. | DOI | MR | Zbl

[43] W. Van Assche. Eigenvalues of Toeplitz matrices associated with orthogonal polynomials. J. Approx. Theory 51 (1987) 360–371. | DOI | MR | Zbl

[44] W. Van Assche. Multiple orthogonal polynomials, irrationality and transcendence. In Continued Fractions: From Analytic Number Theory to Constructive Approximation (Columbia, MO, 1998) 325–342. Contemp. Math. 236. Amer. Math. Soc., Providence, RI, 1999. | MR | Zbl

[45] W. Van Assche. Nearest neighbor recurrence relations for multiple orthogonal polynomials. J. Approx. Theory 163 (2011) 1427–1448. | DOI | MR | Zbl

[46] D. V. Voiculescu, K. J. Dykema and A. Nica. Free Random Variables. CRM Monographs Series 1. Amer. Math. Soc., Providence, RI, 1992. | DOI | MR | Zbl

Cité par Sources :