[Paper Review] Disk counting statistics near hard edges of random normal matrices: the multi-component regime

Yacin Ameur, Christophe Charlier+2 more|arXiv (Cornell University)|Oct 25, 2022

Random Matrices and Applications4 citations

TL;DR

This paper studies disk counting statistics in a two-component Coulomb gas model of random normal matrices with a hard wall separating two disjoint circular regions. It derives exact asymptotics for the moment generating function in the hard edge regime, revealing oscillatory fluctuations governed by Jacobi theta functions and showing that component variances follow Weierstrass ℘-function asymptotics.

ABSTRACT

We consider a two-dimensional point process whose points are separated into two disjoint components by a hard wall, and study the multivariate moment generating function of the corresponding disk counting statistics. We investigate the ``hard edge regime" where all disk boundaries are a distance of order $\frac{1}{n}$ away from the hard wall, where $n$ is the number of points. We prove that as $n o + \infty$, the asymptotics of the moment generating function are of the form \begin{align*} & \exp \bigg(C_{1}n + C_{2}\ln n + C_{3} + \mathcal{F}_{n} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(n^{-\frac{3}{5}})\bigg), \end{align*} and we determine the constants $C_{1},\dots,C_{4}$ explicitly. The oscillatory term $\mathcal{F}_{n}$ is of order $1$ and is given in terms of the Jacobi theta function. Our theorems allow us to derive various precise results on the disk counting function. For example, we prove that the asymptotic fluctuations of the number of points in one component are of order $1$ and are given by an oscillatory discrete Gaussian. Furthermore, the variance of this random variable enjoys asymptotics described by the Weierstrass $\wp$-function.

Motivation & Objective

• To analyze disk counting statistics in a two-component Coulomb gas model of random normal matrices with a hard wall separating two disjoint circular regions.
• To investigate the 'hard edge regime' where disk boundaries are at distance $\mathcal{O}(1/n)$ from the hard wall.
• To derive precise asymptotic expansions for the multivariate moment generating function of the disk counting statistics.
• To characterize the fluctuation behavior of the number of points in each component, particularly their variance and distributional limits.
• To establish connections between the asymptotic fluctuations and special functions such as the Jacobi theta function and Weierstrass ℘-function.

Proposed method

• Formulates a Coulomb gas model with inverse temperature $\beta=2$ and a hard wall at radii $\rho_1$ and $\rho_2$, leading to a point process with two disjoint components.
• Uses the Mittag-Leffler ensemble with a modified potential $Q(z)$ that enforces the hard wall via infinite potential in the annulus $[\rho_1, \rho_2]$.
• Applies balayage techniques to compute the limiting macroscopic measure $\mu_h$, which has singular components on the circles $r=\rho_1$ and $r=\rho_2$.
• Derives the moment generating function of the disk counting statistics and expands it asymptotically as $n \to \infty$ using complex analysis and special function asymptotics.
• Identifies the oscillatory term $\mathcal{F}_n$ in the asymptotic expansion as being expressed via the Jacobi theta function.
• Employs uniform asymptotics of the incomplete gamma function and its connection to the error function and Stirling-type expansions to control error terms.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the moment generating function for disk counting statistics in a two-component Coulomb gas near hard edges?
• RQ2 How do the fluctuations of the number of points in each component behave as $n \to \infty$?
• RQ3 What is the role of the oscillatory term $\mathcal{F}_n$ in the moment generating function, and how is it related to modular forms?
• RQ4 How do the variances of the counts in each component scale asymptotically, and what special function governs this scaling?
• RQ5 Can the joint distribution of counts in the two components be described by a discrete Gaussian with oscillatory corrections?

Key findings

• The moment generating function admits the asymptotic expansion $\exp\big(C_1 n + C_2 \ln n + C_3 + \mathcal{F}_n + \frac{C_4}{\sqrt{n}} + \mathcal{O}(n^{-3/5})\big)$ as $n \to \infty$, with explicit expressions for $C_1, \dots, C_4$.
• The oscillatory term $\mathcal{F}_n$ is of order 1 and is given explicitly in terms of the Jacobi theta function.
• The asymptotic fluctuations of the number of points in one component are of order 1 and follow an oscillatory discrete Gaussian distribution.
• The variance of the number of points in each component is asymptotically described by the Weierstrass $\wp$-function.
• The constants $C_1, C_2, C_3, C_4$ are explicitly computed in terms of $b$, $\alpha$, $\rho_1$, and $\rho_2$.
• The results are derived using uniform asymptotics of the incomplete gamma function and a novel expansion involving singular parts of Laurent series in the $\lambda$-variable.