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 SIGMA, 2016, Volume 12, 093, 18 pages (Mi sigma1175)

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

Peter Eichelsbachera, Thomas Kriecherbauerb, Katharina Schülerb

a Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
b Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany

Abstract: We prove precise deviations results in the sense of Cramér 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., Schüler 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., Schüler K., Venker M., Markov Process. Related Fields 21 (2015), 639–694] have been proved in the dissertation [Schüler K., Ph.D. Thesis, Universität Bayreuth, 2015].

Keywords: determinantal point process; extreme value distribution; Tracy–Widom distribution; moderate deviations; large deviations; superlarge deviations; random matrix theory; Christoffel–Darboux kernel; Riemann–Hilbert problem.

 Funding Agency Grant Number Deutsche Forschungsgemeinschaft SFB/TR 12 All authors acknowledge support received from the Deutsche Forschungsgemeinschaft within the program of the SFB/TR 12.

DOI: https://doi.org/10.3842/SIGMA.2016.093

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ArXiv: 1603.04328
MSC: 60F10; 60B20; 35Q15; 42C05
Received: May 31, 2016; in final form September 11, 2016; Published online September 21, 2016
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Citation: Peter Eichelsbacher, Thomas Kriecherbauer, Katharina Schüler, “Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail”, SIGMA, 12 (2016), 093, 18 pp.

Citation in format AMSBIB
\Bibitem{EicKriSch16} \by Peter~Eichelsbacher, Thomas~Kriecherbauer, Katharina~Sch\"uler \paper Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail \jour SIGMA \yr 2016 \vol 12 \papernumber 093 \totalpages 18 \mathnet{http://mi.mathnet.ru/sigma1175} \crossref{https://doi.org/10.3842/SIGMA.2016.093} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000385016900001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84996537986} 

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This publication is cited in the following articles:
1. T. Kriecherbauer, M. Venker, “Edge statistics for a class of repulsive particle systems”, Probab. Theory Relat. Field, 170:3-4 (2018), 617–655
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