Abstract
We examine the Charlier, Meixner, Krawtchouk and Hahn discrete orthogonal polynomial ensembles, deeply investigated by K. Johansson, using integration by parts for the underlying Markov operators, differential equations on Laplace transforms and moment equations. As for the matrix ensembles, equilibrium measures are described as limits of empirical spectral distributions. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. Factorial moment identities on mean spectral measures may be used towards small deviation inequalities on the rightmost charges at the rate given by the Tracy-Widom asymptotics.
Highlights
Determinantal representations of eigenvalues are the keys for a deep understanding of both the global and local regimes of random matrix and random growth models by means of orthogonal polynomials
Following the strategy of our paper [Le] for families of orthogonal polynomials of the continuous variable, we study spectral limits using the simple tools of integration by parts for the associated Markov generators, differential equations on Laplace transforms and moment identities for the probability densities
In the first part of this work, we describe a general abstract setting to develop integration by parts in the study of the asymptotic properties of mean spectral measures of discrete orthogonal polynomial ensembles through the representation (3)
Summary
Determinantal representations of eigenvalues are the keys for a deep understanding of both the global and local regimes of random matrix and random growth models by means of orthogonal polynomials. Johansson obtained in [Joha1] (see [Se1]) large deviations and fluctuation properties for the rightmost charges of the Meixner orthogonal polynomial ensemble similar to the behavior of the largest eigenvalues of the random matrix models In this line of investigation, we will be interested in this work in the rescaled mean spectral measures μN = E δxi /N (4). With respect to the continuous setting developed in the companion paper [Le] where recurrence equations are used to this task, we work out here (factorial) moment identities for the Charlier, Meixner and Krawtchouk ensembles by the integration by parts formula These moment equations may be used towards small deviation inequalities at fixed size on the rightmost charges max1≤i≤N xi of the Coulomb gas Q associated to these families of orthogonal polynomials, covering by elementary means, the above tail inequalities (9) and (10)
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