Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail

TL;DR

This paper establishes precise deviation results for the maximum of certain determinantal point processes, covering moderate, large, and superlarge deviations, and relates these to the Tracy-Widom law in random matrix theory.

Contribution

It provides the first comprehensive analysis of upper tail deviations for maxima of determinantal point processes, including regimes beyond the Tracy-Widom law.

Findings

Determined leading order tail probabilities for all deviation regimes.

Identified the regime where Tracy-Widom law remains valid.

Extended asymptotic analysis to superlarge deviations with stronger assumptions.

Abstract

We prove precise deviations results in the sense of Cram\'er and Petrov for the upper tail of the distribution of the maximal value for a special class of determinantal point processes that play an important role in random matrix theory. Here we cover all three regimes of moderate, large and superlarge deviations for which we determine the leading order description of the tail probabilities. As a corollary of our results we identify the region within the regime of moderate deviations for which the limiting Tracy-Widom law still predicts the correct leading order behavior. Our proofs use that the determinantal point process is given by the Christoffel-Darboux kernel for an associated family of orthogonal polynomials. The necessary asymptotic information on this kernel has mostly been obtained in [Kriecherbauer T., Schubert K., Sch\"uler K., Venker M., Markov Process. Related Fields 21 (2015), 639-694]. In the superlarge regime these results of do not suffice and we put stronger assumptions on the point processes. The results of the present paper and the relevant parts of [Kriecherbauer T., Schubert K., Sch\"uler K., Venker M., Markov Process. Related Fields 21 (2015), 639-694] have been proved in the dissertation [Sch\"uler K., Ph.D. Thesis, Universit\"at Bayreuth, 2015].