## 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 μ, 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 μ is obtained by a function of independent random variables. Two open problems are also asked.

Original language | English (US) |
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Journal | Journal of Theoretical Probability |

DOIs | |

State | Accepted/In press - 2021 |

### Bibliographical note

Funding Information:Ke Wang was partially supported by Hong Kong RGC Grant GRF 16301618, GRF 16308219 and ECS 26304920.

Funding Information:

The research of Tiefeng Jiang was supported in part by NSF Grant DMS-1209166 and DMS-1406279.

## Keywords

- Eigenvalue
- Gamma distribution
- Gumbel distribution
- Laplace–Beltrami operator
- Plancherel measure
- Random partition
- Restricted Jack measure
- Restricted uniform measure
- Tracy–Widom distribution
- Uniform measure