Statistical Properties of Eigenvalues of Laplace–Beltrami Operators
Abstract We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restric...
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| Autor*in: |
Jiang, Tiefeng [verfasserIn] |
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Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media, LLC, part of Springer Nature 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of theoretical probability - Springer US, 1988, 34(2021), 3 vom: 01. Jan., Seite 1061-1109 |
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| Übergeordnetes Werk: |
volume:34 ; year:2021 ; number:3 ; day:01 ; month:01 ; pages:1061-1109 |
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| DOI / URN: |
10.1007/s10959-020-01061-6 |
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| 520 | |a Abstract We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted uniform measure. Then, by using it together with known results on other three measures, we prove that the global distribution of the eigenvalues is asymptotically a new distribution $$\mu $$, the Gamma distribution, the Gumbel distribution, and the Tracy–Widom distribution, respectively. The Tracy–Widom distribution is obtained for a special case only due to a technical constraint. An explicit representation of $$\mu $$ is obtained by a function of independent random variables. Two open problems are also asked. | ||
| 650 | 4 | |a Laplace–Beltrami operator | |
| 650 | 4 | |a Eigenvalue | |
| 650 | 4 | |a Random partition | |
| 650 | 4 | |a Plancherel measure | |
| 650 | 4 | |a Uniform measure | |
| 650 | 4 | |a Restricted Jack measure | |
| 650 | 4 | |a Restricted uniform measure | |
| 650 | 4 | |a Tracy–Widom distribution | |
| 650 | 4 | |a Gumbel distribution | |
| 650 | 4 | |a Gamma distribution | |
| 700 | 1 | |a Wang, Ke |0 (orcid)0000-0001-8128-3441 |4 aut | |
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10.1007/s10959-020-01061-6 doi (DE-627)OLC2126787834 (DE-He213)s10959-020-01061-6-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Jiang, Tiefeng verfasserin aut Statistical Properties of Eigenvalues of Laplace–Beltrami Operators 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted uniform measure. Then, by using it together with known results on other three measures, we prove that the global distribution of the eigenvalues is asymptotically a new distribution $$\mu $$, the Gamma distribution, the Gumbel distribution, and the Tracy–Widom distribution, respectively. The Tracy–Widom distribution is obtained for a special case only due to a technical constraint. An explicit representation of $$\mu $$ is obtained by a function of independent random variables. Two open problems are also asked. Laplace–Beltrami operator Eigenvalue Random partition Plancherel measure Uniform measure Restricted Jack measure Restricted uniform measure Tracy–Widom distribution Gumbel distribution Gamma distribution Wang, Ke (orcid)0000-0001-8128-3441 aut Enthalten in Journal of theoretical probability Springer US, 1988 34(2021), 3 vom: 01. Jan., Seite 1061-1109 (DE-627)129930903 (DE-600)357043-5 (DE-576)018191886 0894-9840 nnns volume:34 year:2021 number:3 day:01 month:01 pages:1061-1109 https://doi.org/10.1007/s10959-020-01061-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_4027 AR 34 2021 3 01 01 1061-1109 |
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10.1007/s10959-020-01061-6 doi (DE-627)OLC2126787834 (DE-He213)s10959-020-01061-6-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Jiang, Tiefeng verfasserin aut Statistical Properties of Eigenvalues of Laplace–Beltrami Operators 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted uniform measure. Then, by using it together with known results on other three measures, we prove that the global distribution of the eigenvalues is asymptotically a new distribution $$\mu $$, the Gamma distribution, the Gumbel distribution, and the Tracy–Widom distribution, respectively. The Tracy–Widom distribution is obtained for a special case only due to a technical constraint. An explicit representation of $$\mu $$ is obtained by a function of independent random variables. Two open problems are also asked. Laplace–Beltrami operator Eigenvalue Random partition Plancherel measure Uniform measure Restricted Jack measure Restricted uniform measure Tracy–Widom distribution Gumbel distribution Gamma distribution Wang, Ke (orcid)0000-0001-8128-3441 aut Enthalten in Journal of theoretical probability Springer US, 1988 34(2021), 3 vom: 01. Jan., Seite 1061-1109 (DE-627)129930903 (DE-600)357043-5 (DE-576)018191886 0894-9840 nnns volume:34 year:2021 number:3 day:01 month:01 pages:1061-1109 https://doi.org/10.1007/s10959-020-01061-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_4027 AR 34 2021 3 01 01 1061-1109 |
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10.1007/s10959-020-01061-6 doi (DE-627)OLC2126787834 (DE-He213)s10959-020-01061-6-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Jiang, Tiefeng verfasserin aut Statistical Properties of Eigenvalues of Laplace–Beltrami Operators 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted uniform measure. Then, by using it together with known results on other three measures, we prove that the global distribution of the eigenvalues is asymptotically a new distribution $$\mu $$, the Gamma distribution, the Gumbel distribution, and the Tracy–Widom distribution, respectively. The Tracy–Widom distribution is obtained for a special case only due to a technical constraint. An explicit representation of $$\mu $$ is obtained by a function of independent random variables. Two open problems are also asked. Laplace–Beltrami operator Eigenvalue Random partition Plancherel measure Uniform measure Restricted Jack measure Restricted uniform measure Tracy–Widom distribution Gumbel distribution Gamma distribution Wang, Ke (orcid)0000-0001-8128-3441 aut Enthalten in Journal of theoretical probability Springer US, 1988 34(2021), 3 vom: 01. Jan., Seite 1061-1109 (DE-627)129930903 (DE-600)357043-5 (DE-576)018191886 0894-9840 nnns volume:34 year:2021 number:3 day:01 month:01 pages:1061-1109 https://doi.org/10.1007/s10959-020-01061-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_4027 AR 34 2021 3 01 01 1061-1109 |
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10.1007/s10959-020-01061-6 doi (DE-627)OLC2126787834 (DE-He213)s10959-020-01061-6-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Jiang, Tiefeng verfasserin aut Statistical Properties of Eigenvalues of Laplace–Beltrami Operators 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted uniform measure. Then, by using it together with known results on other three measures, we prove that the global distribution of the eigenvalues is asymptotically a new distribution $$\mu $$, the Gamma distribution, the Gumbel distribution, and the Tracy–Widom distribution, respectively. The Tracy–Widom distribution is obtained for a special case only due to a technical constraint. An explicit representation of $$\mu $$ is obtained by a function of independent random variables. Two open problems are also asked. Laplace–Beltrami operator Eigenvalue Random partition Plancherel measure Uniform measure Restricted Jack measure Restricted uniform measure Tracy–Widom distribution Gumbel distribution Gamma distribution Wang, Ke (orcid)0000-0001-8128-3441 aut Enthalten in Journal of theoretical probability Springer US, 1988 34(2021), 3 vom: 01. Jan., Seite 1061-1109 (DE-627)129930903 (DE-600)357043-5 (DE-576)018191886 0894-9840 nnns volume:34 year:2021 number:3 day:01 month:01 pages:1061-1109 https://doi.org/10.1007/s10959-020-01061-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_4027 AR 34 2021 3 01 01 1061-1109 |
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Abstract We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted uniform measure. Then, by using it together with known results on other three measures, we prove that the global distribution of the eigenvalues is asymptotically a new distribution $$\mu $$, the Gamma distribution, the Gumbel distribution, and the Tracy–Widom distribution, respectively. The Tracy–Widom distribution is obtained for a special case only due to a technical constraint. An explicit representation of $$\mu $$ is obtained by a function of independent random variables. Two open problems are also asked. © Springer Science+Business Media, LLC, part of Springer Nature 2021 |
| abstractGer |
Abstract We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted uniform measure. Then, by using it together with known results on other three measures, we prove that the global distribution of the eigenvalues is asymptotically a new distribution $$\mu $$, the Gamma distribution, the Gumbel distribution, and the Tracy–Widom distribution, respectively. The Tracy–Widom distribution is obtained for a special case only due to a technical constraint. An explicit representation of $$\mu $$ is obtained by a function of independent random variables. Two open problems are also asked. © Springer Science+Business Media, LLC, part of Springer Nature 2021 |
| abstract_unstemmed |
Abstract We study the eigenvalues of a Laplace–Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. To study the behaviors of these eigenvalues, we assign partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure, or the Plancherel measure. We first obtain a new limit theorem on the restricted uniform measure. Then, by using it together with known results on other three measures, we prove that the global distribution of the eigenvalues is asymptotically a new distribution $$\mu $$, the Gamma distribution, the Gumbel distribution, and the Tracy–Widom distribution, respectively. The Tracy–Widom distribution is obtained for a special case only due to a technical constraint. An explicit representation of $$\mu $$ is obtained by a function of independent random variables. Two open problems are also asked. © Springer Science+Business Media, LLC, part of Springer Nature 2021 |
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| title_short |
Statistical Properties of Eigenvalues of Laplace–Beltrami Operators |
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https://doi.org/10.1007/s10959-020-01061-6 |
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| author2 |
Wang, Ke |
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Wang, Ke |
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| doi_str |
10.1007/s10959-020-01061-6 |
| up_date |
2024-07-04T08:24:41.870Z |
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