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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| Видавець: | Інститут математики НАН України |
|---|---|
| Дата: | 2016 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2016
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/147855 |
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| Цитувати: | 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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irk-123456789-1478552019-02-17T01:26:53Z Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail Eichelsbacher, P. Kriecherbauer, T. Schüler, K. 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]. 2016 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/147855 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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]. |
| format |
Article |
| author |
Eichelsbacher, P. Kriecherbauer, T. Schüler, K. |
| spellingShingle |
Eichelsbacher, P. Kriecherbauer, T. Schüler, K. Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Eichelsbacher, P. Kriecherbauer, T. Schüler, K. |
| author_sort |
Eichelsbacher, P. |
| title |
Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail |
| 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 |
| publisher |
Інститут математики НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:28:41Z |
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