Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Average characteristic polynomials of determinantal point processes

Adrien Hardy

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Abstract

We investigate the average characteristic polynomial $\mathbb{E}[\prod_{i=1}^{N}(z-x_{i})]$ 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.

Résumé

On s’intéresse au polynôme caractéristique moyen $\mathbb{E}[\prod_{i=1}^{N}(z-x_{i})]$ associé à des variables aléatoires réelles $x_{1},\ldots,x_{N}$ 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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 1 (2015), 283-303.

Dates
First available in Project Euclid: 14 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1421244406

Digital Object Identifier
doi:10.1214/13-AIHP572

Mathematical Reviews number (MathSciNet)
MR3300971

Zentralblatt MATH identifier
1332.60023

Subjects
Primary: 60B10: Convergence of probability measures 60F15: Strong theorems 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05]

Keywords
Determinantal point processes Average characteristic polynomials Strong law of large numbers Random matrices Multiple orthogonal polynomials

Citation

Hardy, Adrien. Average characteristic polynomials of determinantal point processes. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 1, 283--303. doi:10.1214/13-AIHP572. https://projecteuclid.org/euclid.aihp/1421244406.


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