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Partition functions of two-dimensional Coulomb gases with circular root- and jump-type singularities

This paper establishes precise large $n$ asymptotic expansions for the moment generating function of random polynomials formed by eigenvalue moduli of random normal matrices, confirming a conjecture by Byun and Charlier and providing the first free energy expansion for two-dimensional Coulomb gases with circular root- and jump-type singularities.

Original authors: Kohei Noda

Published 2026-10-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kohei Noda

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern physics, there is a quiet but profound effort to understand how countless tiny particles arrange themselves when they are free to move but also push against one another. Imagine a crowded room where everyone wants personal space, yet they are all confined within a specific boundary. This is the world of random normal matrices, a mathematical framework used to model complex systems ranging from the energy levels of heavy atomic nuclei to the behavior of electrons in exotic materials. At the heart of this study lies a concept called the partition function, which acts as a master key. It does not tell us the exact position of every single particle, but rather summarizes the overall statistical behavior of the entire crowd, revealing how likely the system is to be in one state versus another. For decades, scientists have been able to predict the general shape of these crowds, but the details become incredibly difficult to calculate when the particles encounter specific obstacles or irregularities in their environment.

A researcher has now cracked a particularly stubborn version of this problem. They focused on a scenario where the particles are confined to a circular region, but the rules governing their movement change abruptly along a specific ring inside that circle. Think of this ring as a boundary where the rules of the game suddenly shift: on one side, the particles feel a certain kind of resistance, and on the other, they feel a different kind. Furthermore, the strength of this resistance can change in a smooth, gradual way or jump suddenly, creating what mathematicians call "root" and "jump" singularities. These are not physical holes or walls, but rather mathematical features in the potential energy landscape that dictate how the particles interact. Until now, predicting the precise statistical behavior of such a system, especially when the number of particles is very large, was an open challenge. The existing tools worked well for simple cases, but they broke down when faced with these complex, mixed types of irregularities.

The researcher set out to find a precise formula that describes the behavior of this system as the number of particles grows toward infinity. They did not rely on computer simulations or approximations; instead, they derived a rigorous mathematical expansion that predicts the system's properties with high accuracy. Their work confirms a long-standing guess made by other scientists, which suggested that the answer involves a specific family of special functions known as parabolic cylinder functions. These functions are the mathematical equivalent of a universal translator, capable of describing the subtle transitions that occur right at the boundary where the rules change. The researcher successfully proved that these functions are indeed the correct tool for the job, replacing older, less accurate methods that had been used for simpler cases.

What makes this discovery significant is the level of detail it provides. The researcher did not just find a rough estimate; they calculated the first three major terms of the expansion, which correspond to the most dominant effects, the next most important corrections, and the fine-tuning details. This allows for a much sharper picture of the system's behavior than was previously possible. For instance, they were able to determine exactly how the "counting statistics" of the particles behave. This refers to the probability of finding a certain number of particles inside a specific circle. Their formula shows that while the average number of particles scales with the size of the system, the fluctuations around this average follow a very specific pattern that depends on the nature of the irregularities in the potential.

The study also clarifies the limits of their findings. The mathematical derivation holds true for a broad class of smooth, rotationally symmetric potentials, meaning the rules of the system look the same no matter which direction you look. However, the author explicitly notes that their method does not apply to certain power-law potentials that create sharp, cone-like singularities at the very center of the system. These specific cases, which appear in the study of quantum Hall states on surfaces with conical points, require a different mathematical approach that involves more complex functions. By drawing this boundary, the researcher ensures that their results are applied correctly and do not overreach into territory where the current proof does not hold.

In the end, this work provides a new, precise map for navigating the statistical behavior of two-dimensional Coulomb gases with circular irregularities. It bridges the gap between simple, idealized models and the messy, complex realities found in physical systems. By confirming the role of parabolic cylinder functions and providing a detailed expansion of the partition function, the researcher has given the scientific community a powerful new tool. This tool can now be used to predict the behavior of similar systems with greater confidence, potentially aiding in the understanding of phase transitions and the statistical mechanics of disordered systems where particles interact in complex ways. The result is a clearer, more concrete understanding of how order emerges from chaos when the rules of the game change along a circular path.

Technical Summary: Partition functions with circular root- and jump-type singularities

Problem Statement
This paper investigates the precise large- $n$ asymptotic behavior of the partition function $Z_{n,u,a}[Q]$ associated with two-dimensional Coulomb gases (random normal matrices) possessing a radially symmetric external potential $Q(z) = q(|z|)$ . The study focuses on the moment generating function of the random polynomial $p_n(\rho) = \prod_{j=1}^n (|z_j| - \rho)$ , where $\{z_j\}$ are the eigenvalues of the random matrix. Specifically, the author analyzes the quantity:
$E_{n,u,a} := \mathbb{E}\left[ e^{\frac{u}{\pi} \text{Im} \log p_n(\rho)} e^{a \text{Re} \log p_n(\rho)} \right] = \frac{Z_{n,u,a}[Q]}{Z_{n,0,0}[Q]},$
where $\rho > 0$ lies in the bulk of the spectral droplet. The weight function $\omega(z)$ introduces a circular root-type singularity (controlled by parameter $a > -1$ ) and a jump-type singularity (controlled by parameter $u \in \mathbb{R}$ ) along the circle of radius $\rho$ .

Previous work by Byun and Charlier [28] established asymptotic expansions for the case where $a$ is a positive integer, where the coefficients involved associated Hermite polynomials. However, the general case for $a > -1$ remained an open problem, with a conjecture that the asymptotics would involve parabolic cylinder functions instead of Hermite polynomials.

Methodology
The author employs a robust strategy combining the Laplace method, precise Riemann sum approximations, and a decomposition of the logarithmic sum into global and local analysis parts.

Reduction to One-Fold Integrals: Utilizing the Andréief identity, the $n$ -fold integral defining the partition function is reduced to a product of one-fold integrals. The logarithm of the moment generating function is expressed as a sum of logarithms of ratios of these integrals.
Decomposition of the Sum: The sum over indices $j=0, \dots, n-1$ $j = 0, \dots, n - 1$ is split into four parts ( $S_0, S_1, S_2, S_3$ $S_{0}, S_{1}, S_{2}, S_{3}$ ) based on the location of the critical points of the effective potential $V_\tau(r) = q(r) - 2\tau \log r$ $V_{τ} (r) = q (r) - 2 τ lo g r$ relative to the singularity radius $\rho$ $ρ$ .
- $S_0$ : Indices far from the bulk where the critical point is near the origin.
- $S_1$ and $S_3$ : Indices where the critical point is strictly inside or outside the circle of radius $\rho$ , respectively. These are analyzed using the Laplace method and Taylor expansions.
- $S_2$ : The "local" part where the critical point is close to $\rho$ (within a window of size $O(n^{-1/2})$ ). This region requires a detailed local analysis involving parabolic cylinder functions.
Local Analysis: In the critical region $S_2$ , the integrals are approximated using Gaussian integrals. The resulting expressions involve the parabolic cylinder function $D_{-\nu}(x)$ and its derivatives. The author defines a function $H_{a,u}(x)$ involving these functions to capture the local behavior.
Riemann Sum Approximation: The sums $S_1, S_2, S_3$ are approximated by integrals using the Euler-Maclaurin formula. Careful error analysis is performed to ensure the remainder terms are controlled, specifically addressing the divergent terms that arise near the boundaries of the summation intervals.
Cancellation of Divergences: A key technical step involves showing that divergent terms arising in the local ( $S_2$ ) and boundary ( $S_1, S_3$ ) expansions cancel each other out, leaving a well-defined asymptotic expansion.

Key Contributions and Results

Confirmation of the Byun-Charlier Conjecture: The paper confirms the conjecture from [28] that for general $a > -1$ , the large- $n$ asymptotic expansion of the moment generating function involves parabolic cylinder functions rather than associated Hermite polynomials.
Precise Asymptotic Expansion (Theorem 2): The author establishes the following expansion for $\log E_{n,u,a}$ $lo g E_{n, u, a}$ as $n \to \infty$ $n \to \infty$ :
$\log E_{n,u,a} = C_1(u,a)n + C_2(u,a)\sqrt{n} + C_3(u,a) + O\left(\frac{(\log n)^3}{n^{1/12}}\right).$
- $C_1$ and $C_2$ are expressed as integrals involving the potential $Q$ and the function $H_{a,u}$ .
- $C_3$ is derived explicitly in terms of $H_{a,u}$ , the potential $Q$ , and the parameters $\alpha$ and $a$ . The expression for $C_3$ is noted to be simpler than the corresponding term in [28].
Counting Statistics (Theorem 1): As a special case where $a=0$ , the paper provides a refined asymptotic expansion for the disk counting statistics (the moment generating function of the number of eigenvalues inside a disk). This result improves upon previous work [2] by providing the next two terms in the expansion and generalizes results for Mittag-Leffler potentials to a broader class of rotation-invariant potentials.
Free Energy Expansion (Theorem 3): By combining the main result with existing free energy expansions for Coulomb gases [8], the author derives the full asymptotic expansion of the partition function $Z_{n,u,a}[Q]$ , including terms of order $n^2$ , $n \log n$ , $n$ , $\sqrt{n}$ , $\log n$ , and constants. This constitutes the first free energy expansion for 2D Coulomb gases with general circular root- and jump-type singularities.
Cumulants: The paper derives the asymptotic behavior of all cumulants of the counting statistic $N_\rho$ (Corollary 1), generalizing previous results on variance and mean.

Significance and Claims
The author states that this type of asymptotic behavior, characterized by the appearance of parabolic cylinder functions in the coefficients of the expansion, is "completely new in random matrix theory." The work extends the understanding of structured determinants with singularities from one-dimensional point processes (Toeplitz determinants) to two-dimensional Coulomb gases.

The paper claims to provide the first free energy expansion for systems with general circular root- and jump-type singularities. While the error term $O((\log n)^3 n^{-1/12})$ is acknowledged as likely not optimal (with an expectation of $O(n^{-1/2})$ ), the derivation of the precise coefficients $C_1, C_2, C_3$ is presented as a rigorous confirmation of the theoretical conjectures regarding the nature of these singularities. The results are shown to be consistent with and generalize previous findings for specific potentials (like $Q(z)=|z|^{2b}$ ) and integer values of $a$ .

The analysis does not rely on Riemann-Hilbert methods, distinguishing it from some other approaches in the field, and instead utilizes a direct asymptotic analysis of the integrals and sums. The paper also notes that while the assumptions on the potential $Q$ exclude certain power-law behaviors at the origin (conical singularities), the results successfully recover known cases for Mittag-Leffler potentials when formally applied, suggesting a degree of universality in the derived formulas.

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