Average Characteristic Polynomials of Determinantal Point Processes

We investigate the average characteristic polynomial E [∏N i=1(z−xi) ] where the xi’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.