Average characteristic polynomials of determinantal point processes

Hardy, Adrien

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 283-303.

On s’intéresse au polynôme caractéristique moyen $𝔼\left[{\prod }_{i=1}^N(z-x_i)\right]$ 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 $𝔼\left[{\prod }_{i=1}^N(z-x_i)\right]$ 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.

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