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Asymptotics for partition functions of random normal matrix models with singularities

This paper derives explicit asymptotic expansions of partition functions up to order $1$ for determinantal planar Coulomb gases with smooth radial potentials and various jump-type singularities and hard wall regions, thereby unifying and generalizing numerous previous results.

Original authors: Kohei Noda

Published 2026-09-23

📖 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

Imagine a vast, flat stage where countless tiny, invisible particles move freely, repelling one another with a force similar to how magnets push away like poles. In the world of physics, these are not just abstract ideas but real mathematical models used to describe the behavior of electrons in certain materials or the eigenvalues of complex matrices. When these particles are cooled down to a specific state, they settle into a predictable pattern, forming a dense, circular cloud. Scientists have long been interested in the "partition function" of such a system, a single number that acts like a master key, unlocking the statistical secrets of the entire crowd, from how likely the particles are to cluster together to how they fluctuate when disturbed.

However, the real world is rarely perfectly smooth. Sometimes, there are obstacles. Imagine placing a solid, impenetrable wall in the middle of this particle cloud, or carving out a hole where no particles are allowed to exist. This creates a "hard wall," a boundary that forces the particles to rearrange themselves in complex ways. For decades, researchers could only calculate the behavior of these systems in very specific, simple scenarios, such as a single hole in the center or a wall at the very edge. When multiple types of boundaries or irregularities appeared at once, the mathematics became too tangled to solve, leaving a gap in our understanding of how these systems truly behave under stress.

A researcher has now untangled this knot. They developed a unified mathematical framework that describes the behavior of these particle clouds when they face a variety of difficult conditions simultaneously. Their work covers scenarios where the particles are confined by a solid disk in the center, a ring-shaped wall, or a barrier that cuts off the outer edge. Crucially, they also accounted for "singularities," which are points where the rules of the system change abruptly, such as a sudden jump in the force felt by the particles or a sharp change in the density of the crowd. By combining these complex elements, the researcher was able to derive a precise formula that predicts the system's behavior with high accuracy, even as the number of particles grows to be incredibly large.

The core achievement of this study is the ability to predict the "partition function" up to a very fine level of detail. In simpler terms, the researcher calculated not just the broad, average behavior of the particle cloud, but also the subtle, smaller corrections that occur as the system scales up. They found that the presence of a hard wall or a hole introduces specific, predictable shifts in the system's energy and structure. For instance, when a hole is created in the center of the cloud, the particles do not just move away; they adjust their density in a way that creates a distinct, measurable pattern of fluctuation. The researcher's formula captures these adjustments, including a unique, oscillating term that appears only when the particles are trapped between two walls, a phenomenon that had previously been difficult to describe in a general setting.

This work is significant because it unifies many scattered results from previous studies into one coherent picture. Before this, scientists had to use different, often incompatible methods to analyze a single hole versus a ring-shaped barrier. Now, there is a single set of rules that applies to all these situations, whether the particles are near the center, the edge, or squeezed between two boundaries. The researcher proved that their formulas hold true for a wide range of smooth, radial potentials, meaning the results apply to many different physical setups, not just one specific type of material. They also showed how to calculate the probability of finding a large empty region, known as a "hole probability," with unprecedented precision, revealing how the likelihood of such an empty space changes as the system grows.

Beyond just counting particles, the study also sheds light on how the system reacts to smooth, gentle changes. If one were to slightly nudge the particles with a gentle force, the researcher could predict exactly how the average position of the crowd would shift and how much the crowd would jitter. They found that the presence of a hard wall adds a new layer of complexity to these fluctuations, creating a specific type of noise that depends on the shape of the wall and the position of the particles. In some cases, the fluctuations follow a standard bell-curve pattern, but in others, particularly when particles are trapped between two walls, the behavior becomes more exotic, involving a special mathematical function that causes the fluctuations to oscillate in a regular, repeating pattern.

The confidence in these findings is high, as the researcher provided rigorous mathematical proofs for their expansions. They did not rely on computer simulations or approximations that might fail at extreme scales; instead, they derived exact asymptotic expansions, which are formulas that become increasingly accurate as the number of particles increases. Their work confirms that the behavior of these complex systems is governed by a set of universal principles that can be described explicitly. By solving the problem for multiple regimes at once, they have provided a powerful tool for physicists and mathematicians to analyze random matrices and two-dimensional gases, offering a clearer view of how order emerges from chaos even in the presence of sharp boundaries and sudden changes.

Technical Summary: Asymptotics for Partition Functions of Random Normal Matrix Models with Singularities

Problem Statement
This paper investigates the asymptotic behavior of partition functions for two-dimensional Coulomb gases (equivalently, random normal matrix models) at inverse temperature $\beta=2$ . The study focuses on systems governed by a smooth radial potential $Q(z) = q(|z|)$ , subject to three specific types of singularities and constraints:

Hard Wall Regions: The particles are confined to the complement of a forbidden region $U$ , which can be a centered disk ( $D_{\rho_2}$ ), a disk complement ( $D_{\rho_1}^c$ ), or an annulus ( $A_{\rho_1, \rho_2}$ ).
Jump-Type Singularities: The model incorporates counting statistics via moment generating functions with discontinuous weights, allowing for the study of fluctuations in the number of particles within specific radii.
Smooth Perturbations: The system includes smooth radial linear statistics and point-charge insertions (parameterized by $\alpha$ ).

The primary objective is to derive explicit asymptotic expansions of the partition functions (and related quantities like hole probabilities and moment generating functions) up to and including the term of order $O(1)$ as the number of particles $n \to \infty$ .

Methodology
The analysis relies on the radial symmetry of the potential, which allows the partition function to be factorized into a product of one-dimensional radial norms via Andréief's identity. The core technical strategy involves:

Exact Factorization: The deformed partition functions are expressed as products of ratios of radial norms ( $h_{n,j}$ ) and unconstrained norms ( $h_{n,j}$ ), separated into hard-wall ratios and jump deformation factors.
Uniform Asymptotics of Norms: The author performs a detailed asymptotic analysis of the radial norms using the Laplace method and Euler–Maclaurin summation. A critical aspect of the analysis is the uniform treatment of the transition regions where the saddle point of the potential approaches the hard wall boundaries. This involves:
- Bulk Regimes: Standard Laplace expansions.
- Semi-Hard and Hard-Edge Regimes: Expansions involving the complementary error function ( $\text{erfc}$ ) and Airy-type transitions near the boundaries.
- Annular Spectral Gap: Handling the competition between two hard endpoints in the annular case, which leads to oscillatory terms.
Summation Techniques: The sums over the index $j$ (representing the angular momentum modes) are evaluated using the Euler–Maclaurin formula. Special care is taken to match the contributions from the "moderate-deviation" ranges and the "transition" windows (scaling as $n^{-1/2}$ and $n^{-1}$ ) to ensure the cancellation of auxiliary cutoff parameters.
Theta Function Identification: In the annular case, the interaction between the two hard walls generates a bounded correction term expressed via the Jacobi theta function, arising from the discrete nature of the filling fraction in the spectral gap.

Key Contributions and Results
The paper provides a unified framework that recovers and generalizes numerous previous results concerning specific singular regimes (e.g., bulk merging, soft edges, hard edges). The main results are explicit asymptotic expansions for three distinct hard-wall geometries:

General Partition Function Expansions (Theorems 1.2–1.4):
For each geometry (Annulus, Disk Complement, Centered Disk), the logarithm of the partition function is expanded as:
$\log D_n = C_1 n^2 + C_2 n \log n + C_3 n + C_4 \sqrt{n} + C_5 \log n + C_6 + \text{Theta Term} + O(n^{-1/12})$
The coefficients $C_k$ are explicitly computed and decomposed into contributions from the unconstrained model, the hard wall, and the counting statistics insertions.
Hole Probabilities (Theorems 1.6–1.8):
By setting the smooth and counting parameters to zero, the author derives precise large-deviation asymptotics for the probability that a macroscopic region contains no particles.
- For the annular hole, the expansion includes a Jacobi theta term $\log \theta(\xi_n | \tau)$ , reflecting the oscillatory nature of the particle distribution in the gap.
- For the disk complement and centered disk, the expansions involve specific logarithmic corrections ( $-\frac{1}{4}\log n$ and $+\frac{1}{3}\log n$ , respectively) and explicit constants.
Fluctuations of Smooth Linear Statistics (Theorems 1.10–1.12):
The paper establishes the limiting distribution for smooth rotation-invariant linear statistics.
- In the disk complement and centered disk cases, the fluctuations are Gaussian.
- In the annular case, the limiting behavior depends on the difference of the test function at the two walls ( $\delta_\lambda = \lambda(\rho_2) - \lambda(\rho_1)$ ). If $\delta_\lambda = 0$ , the limit is Gaussian. If $\delta_\lambda \neq 0$ , the limit is non-Gaussian due to the additional theta-function contribution, representing a discrete Gaussian mixture.
Multivariate Counting Statistics (Theorem 1.14, Corollary 1.17):
The author derives the joint moment generating function for disk counting statistics in the annular regime. This leads to a multivariate Central Limit Theorem (CLT) with anisotropic scaling ( $\sqrt{n}$ for hard-edge regimes, $n^{1/4}$ for bulk regimes). Crucially, the paper identifies a regime where the counting statistics exhibit order-one discrete Gaussian fluctuations (Corollary 1.17), governed by the fractional part of the filling fraction.

Significance and Claims
The paper claims to provide a unified framework that simultaneously handles general smooth radial potentials, multiple types of singularities (jump and point-charge), and various hard-wall geometries.

Unification: It recovers known results for specific cases (e.g., merging singularities, hard edges, hole probabilities) found in works by Forrester, Ameur, Charlier, Byun, and others, while extending them to more general settings.
Precision: The expansions are derived up to the $O(1)$ term, capturing subtle effects such as the Jacobi theta function in the annular case and specific logarithmic corrections in hole probabilities.
Generality: The results apply to a broad class of potentials satisfying strict subharmonicity and growth conditions, including the Ginibre ensemble and Mittag-Leffler type potentials (though the latter requires $b=1$ for the disk assumption, the framework is general).

The author emphasizes that their approach allows for the derivation of asymptotics for all cumulants of counting statistics and the precise characterization of the transition between Gaussian and non-Gaussian limiting laws in multi-component models. The work does not propose new experimental setups but rather offers a rigorous mathematical foundation for understanding the statistical mechanics of these complex systems.

Technical Summary: Asymptotics for Partition Functions of Random Normal Matrix Models with Singularities

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