Statistical Properties of Eigenvalues of Laplace–Beltrami Operators

We study the eigenvalues of a Laplace–Beltrami operator defined on set symmetric polynomials, where are expressed in terms partitions integers. To behaviors these eigenvalues, we assign with restricted uniform measure, Jack or Plancherel measure. first obtain new limit theorem Then, by using it together known results other three measures, prove that global distribution is asymptotically $$\mu $$ , Gamma distribution, Gumbel and Tracy–Widom respectively. The obtained for special case only due to technical constraint. An explicit representation function independent random variables. Two open problems also asked.