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Disc counting statistics of the real Ginibre ensemble

This paper derives the asymptotic behavior of joint cumulants for real and non-real eigenvalue counts in the real Ginibre ensemble across various regimes, establishing a joint central limit theorem and revealing a universality phenomenon where combined fluctuations match those of complex and symplectic ensembles despite individual dependencies on symmetry classes.

Original authors: Sung-Soo Byun, Yong-Woo Lee

Published 2026-08-25

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CC BY 4.0

Original authors: Sung-Soo Byun, Yong-Woo Lee

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 mathematics, there is a branch dedicated to understanding how things behave when they are thrown together at random. Imagine a collection of points scattered across a surface, where each point's position is determined by a roll of the dice. Scientists call these random point processes, and they appear everywhere, from the distribution of galaxies in the universe to the arrangement of atoms in a material. One of the most fundamental questions researchers ask about these systems is simple: if you draw a circle on the surface, how many points will fall inside it? While the average number of points is easy to predict, the real story lies in the fluctuations—the ways the actual count wiggles up and down around that average. These wiggles reveal deep secrets about how the points interact with one another, telling us whether they are tightly packed, loosely scattered, or following some hidden rule of order.

For decades, mathematicians have studied a specific family of random systems known as the Ginibre ensembles. These are collections of random matrices, which are essentially grids of numbers, where the entries are chosen from a bell-shaped curve of probabilities. The behavior of the numbers inside these grids, called eigenvalues, creates a pattern of points on a flat plane. For two types of these ensembles, the points behave in a very predictable way, forming a uniform cloud that looks like a perfect circle. However, a third type, known as the real Ginibre ensemble, is far more complicated. Because the numbers in its grid are restricted to be real numbers, its points do not just float freely in the plane. Instead, they split into two distinct groups: some points sit directly on a straight line, while the others form pairs that mirror each other across that line. This mixed nature, with points existing in both one dimension and two dimensions simultaneously, has made it incredibly difficult to understand how they fluctuate together.

In a recent study, researchers Sung-Soo Byun and Yong-Woo Lee tackled this difficult problem head-on. They focused on the real Ginibre ensemble and asked a precise question: if you draw a circle in the middle of the plane, how do the numbers of points on the line and the numbers of points in the pairs fluctuate as the size of the grid grows larger and larger? Previous studies had looked at the points on the line and the points in the pairs separately, but no one had successfully described how they behave as a combined team. The authors developed a new mathematical framework to track both groups at once, allowing them to calculate the exact statistical relationship between the two types of points in three different scenarios: when the circle is small and sits deep inside the cloud, when it is right on the edge of the cloud, and when it is large enough to cover the entire cloud and spill over the sides.

What the researchers found was a surprising and elegant resolution to a long-standing puzzle. When they looked at the two groups of points individually, their fluctuations were messy and depended heavily on the specific rules of the system. The points on the line and the paired points seemed to have their own unique personalities, reacting differently to the boundaries of the circle. However, when the researchers added the two groups together to look at the total number of points inside the circle, something remarkable happened. The messy, unique behaviors of the two groups cancelled each other out perfectly. The combined total behaved with a smooth, universal simplicity that was identical to the behavior seen in the other, simpler types of random ensembles. It was as if the two distinct groups, when viewed together, forgot their individual differences and followed a single, unified law of nature.

This discovery provides a complete picture of how these random points behave in the real Ginibre ensemble. The authors proved that while the individual components retain their complex, symmetry-dependent quirks, the system as a whole exhibits a profound universality. They also confirmed a specific prediction about how the variance, or the size of the fluctuations, grows as the system gets larger. The study shows that the fluctuations grow at a rate proportional to the square root of the system size, a result that holds true whether the circle is in the middle of the cloud, on its edge, or outside of it. By solving the problem of counting these points in a disc, the researchers have not only filled a major gap in the theory of random matrices but have also revealed a hidden harmony in a system that previously seemed too complicated to understand. Their work demonstrates that even in a system with mixed dimensions and conflicting behaviors, the whole can be far more orderly and predictable than the sum of its parts.

Technical Summary: Disc Counting Statistics of the Real Ginibre Ensemble

Problem Statement
This paper investigates the counting statistics of eigenvalues for the Real Ginibre Ensemble (GinOE), a model of $N \times N$ random matrices with independent, centered real Gaussian entries. Unlike the Complex (GinUE) and Symplectic (GinSE) Ginibre ensembles, whose spectra consist entirely of non-real eigenvalues, the GinOE spectrum comprises two distinct components: real eigenvalues and pairs of non-real complex-conjugate eigenvalues. This mixed-dimensional structure (one-dimensional real axis and two-dimensional complex plane) fundamentally complicates the analysis of global observables.

The primary objective is to derive the asymptotic behavior of the joint counting statistics for the number of real eigenvalues ( $N^r_N(R)$ ) and non-real eigenvalues ( $N^c_N(R)$ ) contained within a centered disc of radius $R$ . The study covers three macroscopic regimes relative to the circular law (the limiting spectral distribution): the bulk ( $R < 1$ ), the spectral edge ( $R \approx 1$ ), and the exterior ( $R > 1$ ). Specifically, the authors aim to determine the joint cumulants of these counts, their variances, covariances, and limiting distributions as $N \to \infty$ .

Methodology
The authors employ a rigorous analytical approach rooted in the exact finite- $N$ algebraic structure of the GinOE:

Finite- $N$ Determinantal Representation: The analysis begins by leveraging the Pfaffian structure of the GinOE, which reduces to a determinantal representation for even $N$ . The authors derive an explicit formula (Proposition 3.1) for the joint moment generating function of $N^r_N(R)$ and $N^c_N(R)$ in terms of the determinant of a structured matrix. This matrix is decomposed into four components: $L_N(R)$ , $M_N(R)$ , $E_{N,1}(R)$ , and $E_{N,2}(R)$ , whose entries involve incomplete gamma functions and error functions.
Trace Expansion: Using the identity $\log \det(I + X) = \text{Tr} \log(I + X)$ , the joint cumulants are expressed as coefficients in the Taylor expansion of the trace of the logarithm of the structured matrix sum. This reduces the problem to analyzing the asymptotic behavior of trace products of the form $\text{Tr}(\prod \text{structured matrices})$ .
Asymptotic Analysis of Structured Matrices:
- Leading Order Terms: The matrices $L_N(R)$ and $M_N(R)$ are identified as the dominant contributors. The authors establish precise entrywise asymptotic estimates using uniform expansions of the incomplete gamma function and Stirling's approximation.
- Trace Powers: Lemmas 4.4 and 4.5 derive the asymptotic behavior of the pure trace powers $\text{Tr}(L_N(R)^m)$ and $\text{Tr}(M_N(R)^m)$ . These involve integrals of the complementary error function ( $\text{erfc}$ ) and polylogarithm functions, capturing the transition behavior near the spectral edge.
- Error Terms: The matrices $E_{N,1}(R)$ and $E_{N,2}(R)$ are shown to be negligible at the relevant scale ( $O(\sqrt{N})$ ) through localization arguments and exponential decay estimates (Lemma 4.7).
- Mixed Traces: Lemma 4.6 demonstrates that mixed trace products involving only $L_N$ and $M_N$ reduce to the pure trace powers of $M_N$ up to lower-order errors.
Cumulant Extraction: By combining the trace asymptotics with the trace-log expansion, the authors extract the coefficients corresponding to the joint cumulants $\kappa_{j,k}$ for arbitrary orders $j, k$ .

Key Contributions and Results

Joint Cumulant Asymptotics (Theorem 2.2): The paper provides explicit asymptotic formulas for the joint cumulants of $N^r_N(R)$ and $N^c_N(R)$ in all three regimes (bulk, edge, exterior).
- In the bulk ( $R < 1$ ), the joint cumulants scale as $\sqrt{N}$ . The limiting cumulants depend on the radius $R$ and specific constants derived from Stirling numbers of the second kind and polylogarithms.
- In the edge regime ( $R = 1 - s/\sqrt{2N}$ ), the asymptotics involve a scaling function $F(s)$ (defined via an integral of $\text{erfc}$ functions) that describes the crossover.
- In the exterior ( $R > 1$ ), the cumulants for the total count vanish, while the individual components retain specific non-zero limits.
Variance and Covariance Structure (Proposition 2.1):
- The variances of both the real and non-real counts are of order $\sqrt{N}$ , despite the expected number of real eigenvalues being $O(\sqrt{N})$ and non-real being $O(N)$ .
- The covariance between real and non-real counts is negative and of order $\sqrt{N}$ .
- Crucially, the variance of the total eigenvalue count $N_N(R)$ is derived. In the bulk, it scales as $\sqrt{2R}\sqrt{N}$ . In the exterior, it vanishes, indicating a sharp cutoff.
Joint Central Limit Theorem (Corollary 2.3): Since all joint cumulants of order $p \ge 2$ scale as $\sqrt{N}$ , the authors establish a bivariate central limit theorem. The normalized vector $(N^r_N(R), N^c_N(R))$ converges to a Gaussian distribution with a specific covariance matrix $\Sigma_R$ determined by the regime.
Universality Phenomenon: A significant finding is that while the individual counting statistics of real and non-real eigenvalues depend on the symmetry class (GinOE vs. GinUE/GinSE), their combined fluctuations exhibit universality. The limiting cumulants of the total eigenvalue count for the GinOE coincide exactly with those of the GinUE and GinSE in all macroscopic regimes. The symmetry-dependent contributions cancel out precisely when the two species are combined.

Significance and Claims
The paper claims to resolve fundamental open problems regarding the GinOE:

Completing the Picture: It provides the first comprehensive description of the joint fluctuations of the full spectrum, addressing the gap where previous literature focused almost exclusively on real or non-real eigenvalues separately.
Conjecture Confirmation: The results confirm the conjectured asymptotics for the number variance of the total eigenvalue count proposed in recent literature (specifically [3, Conjecture 2.10]).
Edge Asymptotics: The paper establishes the previously missing edge asymptotics for the covariance and variance, completing the description of all macroscopic scaling regimes.
Universality: It reveals a new universality phenomenon where the distinct statistical behaviors of the two spectral components (real vs. non-real) cancel out in the total count, restoring the limiting behavior observed in other symmetry classes.
Technical Advancement: The work extends the analysis of counting statistics to mixed-dimensional point processes, overcoming the lack of radial decomposition that typically simplifies the study of GinUE and GinSE.

The authors note that while the results are derived for centered discs, the methodology relies heavily on the specific structure of the disc (allowing explicit representations via incomplete gamma functions). Extending these precise higher-order cumulant results to general domains remains an open challenge, though the number variance for general domains might be approachable via correlation functions.

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