Correlation functions between singular values and eigenvalues

This paper derives an explicit expression for the joint correlation measure between eigenvalues and singular values in bi-unitarily invariant complex random matrix ensembles, providing closed formulas that simplify significantly for polynomial and Pólya ensembles.

The Gist

Imagine you have a giant, complex machine made of many moving parts. In the world of mathematics, this machine is a Random Matrix—a grid of numbers generated by chance.

To understand how this machine works, mathematicians usually look at it from two different angles:

1. The Eigenvalues: Think of these as the machine's "internal personality." They tell you about the fundamental frequencies or the "vibe" of the system.
2. The Singular Values: Think of these as the machine's "physical strength." They tell you how much the machine stretches or squashes space when it acts on something.

Usually, scientists study these two things separately. It's like studying a person's voice (eigenvalues) and their muscle mass (singular values) as if they had nothing to do with each other. But in reality, they are deeply connected. If you know the muscle mass, you can often guess something about the voice, and vice versa.

The Big Question:
This paper asks: If we know the "muscle mass" (singular values) of a random matrix, what does that tell us about its "personality" (eigenvalues)? Specifically, how do they dance together? Do they repel each other? Do they attract?

The Main Discovery: The "Cross-Covariance" Map

The authors, Matthias Allard and Mario Kieburg, have created a new mathematical map. They call it the 1, k-point correlation function.

Here is a simple analogy:
Imagine you are at a crowded party.

• Singular Values are the guests wearing Red Shirts.
• Eigenvalues are the guests wearing Blue Shirts.

Usually, we count how many Red Shirts are in a room, or how many Blue Shirts are in a room. But this paper asks: If I see a Red Shirt standing at a specific spot, what is the probability of finding a Blue Shirt nearby?

The authors found a formula that acts like a heat map for this party.

• Red Zones (Positive): If the map is red, it means the Red Shirt and Blue Shirt are "attracted." They like to hang out together.
• Blue Zones (Negative): If the map is blue, it means they are "repelled." If a Red Shirt is there, a Blue Shirt is unlikely to be nearby.
• White Zones: They don't care about each other; they are independent.

The "Magic Trick" of the Math

The paper is heavy on advanced calculus, but the core idea relies on a "magic trick" called a bijection.

Think of the singular values and eigenvalues as two different languages describing the same story. The authors found a dictionary (a mathematical formula) that translates the story from "Singular Value Language" directly into "Eigenvalue Language."

However, there's a catch. The story has a strict rule: The product of all the Red Shirts must equal the product of all the Blue Shirts. (This is a mathematical law called the determinant identity). Because of this rule, you can't just translate them independently; they are locked together.

The authors figured out how to unlock this relationship for finite-sized machines (matrices that aren't infinitely big). They derived a specific formula that tells you exactly how the "Red" and "Blue" guests influence each other's positions.

Special Cases: The "Polynomial" and "Pólya" Parties

The math gets even simpler if the random matrix belongs to a special club called a Polynomial Ensemble or a Pólya Ensemble.

• Analogy: Imagine the party guests aren't just random; they are wearing outfits that follow a specific pattern (like a uniform or a matching set).
• The Result: When the matrix is from this special club, the complex "heat map" formula simplifies drastically. It becomes a clean, neat equation that uses a "kernel" (a master key) to describe the whole relationship.

This is huge because it allows scientists to calculate these relationships for famous types of matrices (like the Laguerre and Jacobi ensembles) without getting lost in a sea of numbers.

Why Does This Matter?

You might ask, "Who cares about random grids of numbers?"

1. Quantum Chaos: In the physics of chaotic systems (like the inside of a black hole or a chaotic atom), the "singular values" often describe the energy, while "eigenvalues" describe the stability. Knowing how they correlate helps physicists predict how these systems behave.
2. Data Science: In machine learning, large matrices represent data. Understanding the link between the "strength" of the data (singular values) and its "structure" (eigenvalues) can help improve algorithms for recognizing patterns.
3. The "Single Ring" Theorem: There is a famous theory that says if you have a huge random matrix, the eigenvalues form a ring shape. This paper helps us understand the corrections to that ring when the matrix isn't infinitely big, giving us a more precise picture of reality.

The Visual Proof

The paper includes some beautiful contour plots (Figure 1). Imagine a topographic map of a mountain range.

• The "mountains" (positive values) show where the eigenvalues and singular values like to cluster together.
• The "valleys" (negative values) show where they avoid each other.
• The authors show that for small matrices, this map is very jagged and complex. But as the matrix gets bigger (like going from 3x3 to 25x25), the map smooths out, revealing a universal pattern that is the same for different types of matrices.

In a Nutshell

This paper is like discovering a new law of gravity for the quantum world. It provides the first precise, finite-sized formula that connects the "shape" of a random system (singular values) to its "soul" (eigenvalues). It turns a messy, complicated relationship into a clear, calculable map, allowing scientists to predict how these two fundamental properties of randomness interact.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in Random Matrix Theory (RMT): the lack of explicit formulas describing the joint statistical correlations between the singular values and eigenvalues of finite-dimensional complex random matrices.

• Context: While the statistics of singular values (via Singular Value Decomposition, SVD) and eigenvalues (via Schur decomposition) are well-studied individually, they are rarely analyzed together.
• Motivation: In applications such as Quantum Chaos, Quantum Chromodynamics, and Time Series Analysis, both spectra are often complementary. Existing results (e.g., the Haagerup-Larson theorem and the Single Ring theorem) mostly describe the limiting behavior as matrix size $n \to \infty$ or relate the marginal densities only.
• Specific Challenge: For finite matrix size $n$, the relationship between eigenvalues $z$ and singular values $\sigma$ is constrained by the identity $|\det(X)| = \prod |z_k| = \prod \sigma_k$. This constraint, combined with Weyl's inequalities, implies non-trivial correlations. The authors aim to derive the joint probability measure (specifically the $j, k$-point correlation functions) for $j$ eigenvalues and $k$ singular values for finite $n$, under the assumption of bi-unitary invariance.

2. Methodology

The authors employ a rigorous framework based on Harmonic Analysis and Integral Transforms to bridge the gap between the two spectral types.

• Bi-unitary Invariance: The study focuses on ensembles where the probability density $f_G(X)$ is invariant under $X \to U_1 X U_2$ for unitary matrices $U_1, U_2$. This allows the ensemble to be identified with the distribution of its singular values.
• The SEV Transform: The core tool is the bijection established in previous work [32] between the joint density of squared singular values ($f_{SV}$) and the joint density of eigenvalues ($f_{EV}$). This transform involves the Spherical Transform ($S$) and the Mellin Transform ($M$).
• Correlation Measures: The authors define the $j, k$-point correlation measure $\mu_{j,k}$ weakly via test functions. They specifically target the $1, k$-point correlation function ($f_{1,k}$), which describes the joint density of one squared eigenradius ($r = |z|^2$) and $k$ squared singular values ($a = \sigma^2$).
• Ensemble Classes:
  • Polynomial Ensembles: Distributions where the singular value density is a determinantal point process with a specific kernel structure.
  • Pólya Ensembles: A subclass of polynomial ensembles where the weight function $w$ allows for specific derivative structures, leading to further simplifications.

3. Key Contributions and Results

A. General Formula for $1, k$-Point Correlation (Theorem 1.1)

For a general bi-unitarily invariant ensemble with squared singular value density $f_{SV}$, the authors derive an explicit integral representation for the $1, k$-point correlation function $f_{1,k}(r; a_1, \dots, a_k)$.

• The formula involves a sum over indices $j$, complex contour integrals over $s$, and a determinant involving the Vandermonde structure $\Delta_n$.
• Key Insight: Despite the joint measure of all eigenvalues and singular values potentially being singular (due to the determinant constraint), the marginal measure for one eigenradius and $k$ singular values is a well-defined density function for $n > 1$.
• Special Cases: Explicit formulas are provided for $n=1$ (Dirac delta) and $n=2$ (Proposition 2.7), serving as a consistency check.

B. Conditional Level Density (Corollary 1.2)

The paper provides a closed-form expression for the conditional level density of squared eigenradii given fixed squared singular values. This is derived by dividing the joint density by the singular value density, resulting in a formula involving determinants of Heaviside step functions and powers of ratios $a_b/r$.

C. Simplification for Polynomial Ensembles (Theorem 1.5)

When the singular values follow a Polynomial Ensemble (a determinantal point process with kernel $K$), the general formula simplifies drastically.

• The $1, k$-point function is expressed in terms of the correlation kernel $K$ of the singular values.
• The result is given as a determinant of size $(k+1) \times (k+1)$ involving integrals of the kernel $K$ against specific weight functions $\phi$ and $\Omega$.
• This reveals that the interaction between eigenvalues and singular values is entirely encoded in the kernel $K$ of the singular value statistics.

D. Pólya Ensembles and Cross-Covariance (Proposition 1.7 & Corollary 4.2)

For Pólya Ensembles (which include classical Laguerre and Jacobi ensembles), the authors derive a computationally efficient formula for the cross-covariance density ($cov_{1,k}$).

• The cross-covariance is defined as $f_{1,k} - f_{1,0}f_{0,k}$, quantifying the deviation from independence.
• The formula involves incomplete Mellin transforms of the weight function $w$ and its derivatives.
• Regularity Analysis (Corollary 1.8): The authors prove that for Pólya ensembles, the $1, k$-point correlation function is $C^{n-3}$ continuous but has a discontinuity in its $(n-2)$-th derivative along the line $r = a_j$. This highlights a specific singularity structure arising from the boundary conditions of the spectra.

E. Explicit Examples

The theory is applied to:

1. Laguerre Ensemble (Ginibre): Weight $w(x) = x^\alpha e^{-x}$.
2. Jacobi Ensemble (Truncated Unitary): Weight $w(x) = x^\alpha (1-x)^\beta$.
   Explicit expressions for the cross-covariance are derived using hypergeometric functions and incomplete Gamma/Beta functions.

4. Significance and Implications

• Finite-Size Theory: The paper provides the first explicit finite-$n$ formulas linking eigenvalue and singular value statistics, acting as a finite-dimensional counterpart to the Haagerup-Larson theorem.
• New Statistical Tool: The introduction of the cross-covariance density offers a new way to quantify the "repulsion" or "attraction" between eigenradii and singular values.
  • Physical Interpretation: Negative regions in the cross-covariance plots indicate repulsion (variables are less likely to co-occur than if independent), while positive regions indicate attraction.
• Universality and Scaling: Numerical contour plots (Figure 1) suggest that non-trivial spectral statistics emerge at the "hard edge" (origin) and appear universal across different ensembles (Laguerre vs. Jacobi) after appropriate scaling, even for moderate matrix sizes ($n=25$).
• Future Directions: The results lay the groundwork for studying the $n \to \infty$ limit of these cross-correlations, potentially refining the Single Ring Theorem and the Haagerup-Larson theorem by providing finite-size corrections. It also opens the door to studying perturbations of bi-unitary invariance.

Summary

Allard and Kieburg successfully bridge the gap between singular value and eigenvalue statistics for finite random matrices. By leveraging harmonic analysis and the specific structure of polynomial and Pólya ensembles, they derive exact formulas for joint correlation functions and cross-covariances. These results not only generalize known limiting theorems to finite dimensions but also provide a detailed map of the intricate correlations between the two spectral types, revealing universal behaviors at the spectral edges.