[Paper Review] Statistical properties of eigenvalues of Laplace-Beltrami operators

Tiefeng Jiang, Ke Wang|arXiv (Cornell University)|Feb 1, 2016

Random Matrices and Applications31 references3 citations

TL;DR

This paper studies the statistical distribution of eigenvalues of the Laplace-Beltrami operator on symmetric polynomials, parameterized by integer partitions. By assigning random partitions via four measures—restricted uniform, restricted Jack, uniform, and Plancherel—it establishes that the eigenvalues asymptotically follow a new distribution $\mu$, the Gamma, Gumbel, or Tracy-Widom distribution, with the Tracy-Widom limit proven only for $\alpha=1$ due to technical constraints.

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. By assigning partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure or the Plancherel measure, 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. An explicit representation of $μ$ is obtained by a function of independent random variables. We also derive an independent result on random partitions itself: a law of large numbers for the restricted uniform measure. Two open problems are also asked.

Motivation & Objective

• To understand the global statistical behavior of eigenvalues $\lambda_\kappa$ of the Laplace-Beltrami operator $\Delta_\alpha$ defined on symmetric polynomials.
• To analyze how these eigenvalues behave when the partition $\kappa$ is sampled under different probability measures: restricted uniform, restricted Jack, uniform, and Plancherel.
• To derive the asymptotic distribution of $\lambda_\kappa$ under each measure, particularly identifying limiting laws in the large $n$ limit.
• To establish a new limit theorem for the restricted uniform measure and use it to unify results across all four measures.
• To identify the limiting distribution $\mu$ as an explicit function of independent random variables and pose two open problems.

Proposed method

• The eigenvalues $\lambda_\kappa$ are expressed in terms of partitions $\kappa$ via the formula $\lambda_\kappa = n(m-1) + a(\kappa')\alpha - a(\kappa)$, where $a(\kappa)$ is a combinatorial function of the partition.
• The restricted uniform measure assigns equal probability to all partitions of $n$ with at most $m$ parts, enabling analysis of eigenvalue fluctuations under this constraint.
• The restricted Jack measure is used to study eigenvalues under $\alpha$-deformed symmetric functions, linking to Jack polynomials and orthogonal polynomial theory.
• The uniform and Plancherel measures are applied in the infinite-variable limit ($m \to \infty$), where $\kappa$ is sampled uniformly or according to the Plancherel measure on integer partitions of $n$.
• The limiting distribution of $\frac{2}{\alpha} \cdot \frac{\lambda_\kappa}{n^2}$ is shown to converge to the distribution of $\sum_{i=1}^m Y_i^2$, where $Y_i$ are uniformly distributed on the simplex $\sum y_i = 1$, $y_i \in [0,1]^m$.
• Geometric probability techniques are used to compute the cumulative distribution functions and densities of the limiting laws, particularly for $m=2$ and $m=3$, via volume integrals over the simplex intersected with $\ell^2$-balls.

Experimental results

Research questions

• RQ1 What is the asymptotic distribution of the eigenvalues $\lambda_\kappa$ when the partition $\kappa$ is sampled under the restricted uniform measure?
• RQ2 How do the eigenvalues behave under the restricted Jack measure, and what is the limiting distribution in the large $n$ limit?
• RQ3 What is the global limiting distribution of $\lambda_\kappa$ under the uniform and Plancherel measures as $n \to \infty$?
• RQ4 Can the Tracy-Widom distribution emerge as a limiting law for $\lambda_\kappa$, and under what conditions?
• RQ5 What is the explicit representation of the new limiting distribution $\mu$, and how is it related to independent random variables?

Key findings

• Under the restricted uniform measure, a new limit theorem is established, showing that the normalized eigenvalues converge to a non-Gaussian limiting distribution $\mu$.
• For the restricted Jack measure, the eigenvalue distribution asymptotically converges to the Gamma distribution.
• Under the uniform measure, the eigenvalue distribution converges to the Gumbel distribution in the large $n$ limit.
• Under the Plancherel measure, the eigenvalue distribution converges to the Tracy-Widom distribution, but this result is proven only for $\alpha = 1$ due to technical constraints.
• The limiting distribution $\mu$ is explicitly represented as a function of independent random variables, providing a new analytical characterization.
• For $m=2$, the limiting law of $\frac{2}{\alpha} \cdot \frac{\lambda_\kappa}{n^2}$ has a density $f(t) = \frac{1}{\sqrt{2t-1}}$ on $[\frac{1}{2}, 1]$.