Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail
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 superlarg...
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| Date: | 2016 |
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Інститут математики НАН України
2016
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| Cite this: | Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail / P. Eichelsbacher, T. Kriecherbauer, K. Schüler // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 39 назв. — англ. |
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Eichelsbacher, P. Kriecherbauer, T. Schüler, K. 2019-02-16T09:20:07Z 2019-02-16T09:20:07Z 2016 Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail / P. Eichelsbacher, T. Kriecherbauer, K. Schüler // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 60F10; 60B20; 35Q15; 42C05 DOI:10.3842/SIGMA.2016.094 https://nasplib.isofts.kiev.ua/handle/123456789/147855 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]. This paper is a contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy. The full collection is available at http://www.emis.de/journals/SIGMA/Deift-Tracy.html. All authors acknowledge support received from the Deutsche Forschungsgemeinschaft within the program of the SFB/TR 12. The second author is grateful to Dr. Martin Venker for many fruitful discussions while collaborating on [28] that have influenced the presentation of the present paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail |
| spellingShingle |
Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail Eichelsbacher, P. Kriecherbauer, T. Schüler, K. |
| title_short |
Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail |
| title_full |
Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail |
| title_fullStr |
Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail |
| title_full_unstemmed |
Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail |
| title_sort |
precise deviations results for the maxima of some determinantal point processes: the upper tail |
| author |
Eichelsbacher, P. Kriecherbauer, T. Schüler, K. |
| author_facet |
Eichelsbacher, P. Kriecherbauer, T. Schüler, K. |
| publishDate |
2016 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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].
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147855 |
| citation_txt |
Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail / P. Eichelsbacher, T. Kriecherbauer, K. Schüler // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 39 назв. — англ. |
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2025-12-02T04:27:09Z |
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