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Finite size corrections for real eigenvalues of the elliptic Ginibre matrices

This paper derives finite size corrections for the real eigenvalue densities of elliptic Ginibre matrices in the orthogonal symmetry class across global and edge scaling regimes under both strong and weak non-Hermiticity, thereby extending previous asymptotic findings and recovering known results for the Gaussian orthogonal ensemble in the Hermitian limit.

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

Published 2026-08-26

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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 large collections of random numbers behave when arranged into grids, known as matrices. These grids are not just abstract puzzles; they serve as powerful models for the physical world, helping scientists describe everything from the energy levels of heavy atoms to the fluctuations in financial markets. A central question in this field is how the numbers inside these grids, called eigenvalues, distribute themselves. In many systems, these numbers settle into predictable patterns, much like how gas molecules spread out to fill a container. However, when the numbers in the grid are real rather than complex, a unique phenomenon occurs: some of these eigenvalues are forced to stay on the real number line, while others pair up in the complex plane. This creates a mixed system that is harder to predict and offers a bridge between two major theories of randomness: one that describes perfectly symmetric systems and another that describes chaotic, asymmetric ones.

Scientists have long known the broad shape of these distributions when the grids are infinitely large. But in the real world, systems are never infinite; they have a finite size. Just as a small drop of water behaves differently than a vast ocean, a matrix with a few hundred numbers behaves differently than one with billions. The subtle differences that arise from this finite size are called corrections, and understanding them is crucial for making precise predictions in physics and engineering. For decades, researchers could only describe the main, large-scale shape of these distributions. The finer details, the small ripples that appear when the system is not yet infinite, remained elusive, particularly for a specific type of random matrix that smoothly transitions between order and chaos.

A recent study by Sung-Soo Byun and Yong-Woo Lee has finally mapped these hidden ripples with high precision. The researchers focused on a specific family of matrices known as the elliptic Ginibre ensemble. Imagine a dial that controls the nature of the matrix: at one end, the matrix is completely random and asymmetric; at the other, it is perfectly symmetric. By turning this dial, the matrix morphs from one state to the other. The authors wanted to know exactly how the density of the real eigenvalues changes as the matrix grows larger, specifically looking at the small corrections that appear before the system reaches its infinite limit. They examined this process in two distinct ways: first, when the matrix is held at a fixed level of asymmetry, and second, when the matrix is slowly becoming more symmetric as it grows.

The team discovered that the behavior of these real eigenvalues is remarkably consistent, yet subtly dependent on the specific conditions of the system. In the regime where the matrix remains distinctly asymmetric, they found that the density of real eigenvalues follows a simple, uniform pattern across the allowed range, but with a tiny, predictable correction that fades away exponentially fast as the matrix gets bigger. This correction is so small that it vanishes almost entirely, leaving behind a clean, flat distribution. However, as the matrix approaches the symmetric state, the story becomes more intricate. Here, the distribution of eigenvalues begins to curve, shifting from a flat line to a shape that resembles a semi-circle, a classic pattern known as Wigner's semi-circle law. The researchers calculated the exact mathematical form of the transition, showing how the system interpolates between these two shapes.

Perhaps the most significant finding concerns the edges of the distribution, where the eigenvalues are most likely to be found. In the asymmetric case, the edge behaves in a universal way, meaning its shape is the same regardless of the specific details of the matrix, a hallmark of deep physical laws. However, the researchers found that the tiny corrections to this edge are not universal; they depend on the specific settings of the matrix. This distinction is vital because it tells scientists that while the broad strokes of the pattern are fixed by nature, the fine details carry the signature of the specific system being studied. In the symmetric limit, their results perfectly matched previous findings for a well-known class of matrices, confirming the accuracy of their new methods.

By providing these precise formulas, the authors have given scientists a new tool to measure how quickly a finite system settles into its universal behavior. They showed that the speed of this convergence depends on whether the system is in a strongly asymmetric state or a weakly asymmetric one. In the former, the system settles very quickly, with errors disappearing at a rapid rate. In the latter, the convergence is slower and more complex, involving a delicate interplay of factors that the authors have now explicitly described. This work does not just fill a gap in the mathematical record; it offers a clearer lens through which to view the transition between order and chaos in random systems, ensuring that when scientists model real-world phenomena with finite data, they have the most accurate description of the underlying randomness possible.

Technical Summary: Finite Size Corrections for Real Eigenvalues of the Elliptic Ginibre Matrices

Problem Statement
This work investigates the finite size corrections of the real eigenvalue densities for the elliptic Ginibre orthogonal ensemble (elliptic GinOE). The elliptic GinOE serves as an interpolation between the real Ginibre ensemble (GinOE, $\tau=0$ ) and the Gaussian orthogonal ensemble (GOE, $\tau=1$ ), governed by a parameter $\tau \in [0, 1]$ . While the leading-order asymptotic behaviors of the expected number of real eigenvalues and their densities are known in various scaling limits, this paper addresses the subleading terms—specifically the finite size corrections—in both the global and edge scaling regimes. The study focuses on the rate of convergence to universal laws and the precise statistical properties of real eigenvalues, which are non-trivial due to the real symmetry class allowing for real eigenvalues with non-zero probability.

Methodology
The authors employ a rigorous analytical approach based on the skew-orthogonal polynomial representation of the real eigenvalue density. The 1-point function $R_N(x)$ , which characterizes the density of real eigenvalues, is decomposed into two components, $R_N^1(x)$ and $R_N^2(x)$ , expressed in terms of classical Hermite polynomials $H_k(x)$ .

The core of the analysis relies on the Plancherel-Rotach asymptotic formulas for Hermite polynomials. The authors categorize the analysis into distinct regimes based on the location of $x$ relative to the foci of the limiting elliptic domain ( $\pm 2\sqrt{\tau}$ ):

Oscillatory Regime: $|x| < 2\sqrt{\tau}$ .
Critical Regime: $|x \pm 2\sqrt{\tau}| = O(N^{-2/3})$ .
Exponential Regime: $|x| > 2\sqrt{\tau}$ .

To handle the summation in the expression for $R_N^1(x)$ , the authors utilize an integral representation derived from a generalized Christoffel-Darboux formula, avoiding direct summation. They apply asymptotic expansions of the Hermite polynomials (including higher-order terms up to $O(N^{-1})$ or $O(N^{-2/3})$ ) within these integral representations. The analysis distinguishes between two primary scaling limits for the non-Hermiticity parameter $\tau$ :

Strong Non-Hermiticity: $\tau$ is fixed in $[0, 1)$ .
Weak Non-Hermiticity: $\tau \to 1$ as $N \to \infty$ , specifically $\tau = 1 - \alpha^2/N$ (bulk scaling) and $\tau = 1 - \alpha^2/N^{1/3}$ (edge scaling).

The proofs involve careful estimation of oscillatory integrals using integration by parts and the evaluation of initial terms via hypergeometric functions and Stirling's approximation.

Key Contributions and Results

Global Density Corrections:
- Strong Non-Hermiticity ( $\tau$ fixed): The authors derive the finite size correction for the 1-point function $R_N(x)$ . They show that $R_N(x) = R_s^{(0)}(x)N^{1/2} + O(e^{-\epsilon N})$ , where the leading term is uniform. Consequently, the normalized density $\rho_N(x)$ admits an expansion $\rho_N(x) = \rho_s^{(0)}(x) + \rho_s^{(1)}(x)N^{-1/2} + O(N^{-1})$ . The leading term $\rho_s^{(0)}(x)$ is uniform, while the correction term $\rho_s^{(1)}(x)$ is a constant independent of $x$ .
- Weak Non-Hermiticity ( $\tau = 1 - \alpha^2/N$ ): The paper provides the first two terms of the asymptotic expansion for $R_N(x)$ in the bulk: $R_N(x) = R_w^{(0)}(x)N + R_w^{(1)}(x) + O(N^{-1})$ . The leading term $R_w^{(0)}(x)$ involves the error function and interpolates between the uniform distribution and the Wigner semi-circle law. The correction term $R_w^{(1)}(x)$ is explicitly calculated, involving error functions and exponential terms dependent on $\alpha$ and $x$ .
Edge Scaling Corrections:
- Strong Non-Hermiticity: The edge scaling limit is analyzed near $x = 1+\tau$ . The rescaled density $R_N^s(\xi)$ is shown to converge to a limit involving the error function and the Airy function structure, with a subleading correction of order $N^{-1/2}$ . The leading order term is universal (independent of $\tau$ ), while the correction term depends on $\tau$ .
- Weak Non-Hermiticity: In the regime $\tau = 1 - \alpha^2/N^{1/3}$ , the edge scaling limit involves a non-Hermitian extension of the Airy point process, denoted by $Ai_\alpha(\xi)$ . The authors derive the expansion $R_N^w(\xi) = R_w^{(0)}(\xi) + R_w^{(1)}(\xi)N^{-1/3} + O(N^{-2/3+\epsilon})$ . The leading term $R_w^{(0)}$ depends on $\alpha$ , reflecting the curvature of the ellipse, while the correction term $R_w^{(1)}$ is given by a complex expression involving $Ai_\alpha$ and its derivative.
Recovery of Known Results:
- In the limit $\tau \to 1$ (GOE), the derived finite size corrections for the global density recover the results established by Forrester, Frankel, and Garoni.
- For the edge scaling in the GOE limit ( $\alpha=0$ ), the correction term is shown to be expressible as a derivative of the leading term, implying a re-centering of the scaling variable. This confirms the optimal convergence rate of $O(N^{-2/3})$ for the GOE edge, correcting a minor typo in a previous work by Forrester et al.

Significance
The paper claims to extend recent findings on the elliptic GinOE by providing the precise rates of convergence (finite size corrections) for real eigenvalue densities. These corrections are significant because they encompass important statistical properties, such as counting statistics, which have physical implications in systems like two-species particle plasmas and annihilating Brownian motions. By recovering the known finite size corrections for the GOE and providing new results for the intermediate elliptic regimes, the work bridges the gap between non-Hermitian random matrix theory and Hermitian universality classes. The results demonstrate that while the leading order edge scaling limit is universal in the strong non-Hermitian regime, the subleading corrections are non-universal and depend on the parameter $\tau$ . In the weak non-Hermitian regime, the leading order itself depends on the parameter $\alpha$ , indicating a change in the universality class determined by the geometry of the eigenvalue support.

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