Real Ginibre Spherical Ensemble Analysis

• The real Ginibre spherical ensemble is a non-Hermitian random matrix model defined by matrices of the form A B⁻¹, featuring both real eigenvalues and complex-conjugate pairs.
• Its eigenvalue distribution is derived using the real Schur decomposition and a Pfaffian point process framework, with computations rooted in skew-orthogonal polynomials and stereographic projection.
• The ensemble exhibits distinct asymptotic regimes—from large deviations at the N² scale and intermediate saddle-point behavior at √N scale to local Gaussian fluctuations near the mean real eigenvalue count.

Searching arXiv for the cited papers to ground the article in current literature. Searching arXiv for (Forrester, 6 Aug 2025, Mays, 2012), and (Alishahi et al., 2014). arxiv_search query="(Forrester, 6 Aug 2025)" The real Ginibre spherical ensemble is a non-Hermitian random matrix ensemble built from two independent standard real Gaussian $N\times N$ matrices, with recent asymptotic work formulating the model as $A B^{-1}$ and earlier treatments of the real spherical ensemble writing $Y=A^{-1}B$. Its spectrum contains both real eigenvalues and complex-conjugate pairs, and the central counting observable is $p_{N,M}^{\rm r}$, the probability that exactly $M$ eigenvalues are real, with $M$ constrained to have the same parity as $N$. The ensemble sits at the intersection of generalized eigenvalue problems, spherical geometry, Pfaffian point processes, and Coulomb gas asymptotics; its most distinctive feature is that the expected number of real eigenvalues grows only on the order of $\sqrt N$, while atypical counts exhibit distinct large-, intermediate-, and local-deviation regimes (Forrester, 6 Aug 2025).

1. Definition and geometric formulation

In the asymptotic study of real eigenvalue counts, the real Ginibre spherical ensemble consists of random matrices of the form

$A B^{-1},$

where $A$ and $A B^{-1}$0 are independent standard real Gaussian $A B^{-1}$1 matrices. In earlier systematic work on the real spherical ensemble, the same class is written as

$A B^{-1}$2

and interpreted as the generalized eigenvalue problem

$A B^{-1}$3

This generalized-eigenvalue formulation is central because it makes the spectral geometry explicit: the eigenvalues are naturally analyzed after a fractional linear transformation that maps the relevant planar domain to disk or spherical coordinates (Mays, 2012).

A standard coordinate change is

$A B^{-1}$4

which maps the upper half-plane to the unit disk, while points on the real axis are sent to the unit circle. In these variables, the induced eigenvalue probability density is rotationally invariant. The geometric interpretation is that stereographic projection converts the spectral problem into one on the sphere, with real eigenvalues associated to a distinguished great circle and nonreal eigenvalues occupying the complementary two-dimensional region (Mays, 2012).

This geometric setting places the ensemble within the “sphere” branch of the plane–sphere–anti-sphere trichotomy for non-Hermitian random matrices. The plane is associated with Ginibre-type iid matrices, the sphere with matrices of the form $A B^{-1}$5, and the anti-sphere with truncations of orthogonal or unitary matrices. A plausible implication is that the real Ginibre spherical ensemble should be read not merely as a deformation of Ginibre statistics, but as the curved-space representative of the same non-Hermitian symmetry class (Mays, 2012).

2. Matrix density, eigenvalue structure, and Pfaffian formalism

For the real spherical ensemble, the matrix-element density takes the explicit form

$A B^{-1}$6

which is, up to constants, a matrix version of the Cauchy (Lorentzian) distribution. The passage from matrix entries to eigenvalues is carried out through the real Schur decomposition

$A B^{-1}$7

with $A B^{-1}$8 orthogonal and $A B^{-1}$9 block upper-triangular, containing real eigenvalues on the diagonal and $Y=A^{-1}B$0 blocks for complex-conjugate pairs (Mays, 2012).

After the change of variables and the integrations over orthogonal and triangular degrees of freedom, the eigenvalue joint density separates into factors associated with real eigenvalues and complex pairs, together with a Vandermonde-like product over all eigenvalues. The resulting point process is Pfaffian rather than determinantal. The $Y=A^{-1}B$1-point correlation functions have the structure

$Y=A^{-1}B$2

where $Y=A^{-1}B$3 is a $Y=A^{-1}B$4 matrix kernel. This is the characteristic $Y=A^{-1}B$5 structure shared with the real Ginibre ensemble and sharply distinguishes the real spherical model from the complex spherical ensemble, whose correlations are determinantal (Mays, 2012).

The standard computational framework is the five-step method based on (skew-)orthogonal polynomials. In sphere coordinates, the skew-orthogonal polynomials simplify substantially; in particular, monomials may be used as the basic building blocks. For even $Y=A^{-1}B$6, the generating function for the probabilities of observing a specified number of real eigenvalues can be written as

$Y=A^{-1}B$7

with explicit coefficients involving gamma functions. This exact product structure is the starting point for the later steepest-descent analysis of $Y=A^{-1}B$8 (Mays, 2012).

Odd matrix size requires additional care because Pfaffian reductions are naturally even-dimensional. Two complementary approaches are used in the literature: extension of the even case by adding a row and column, and a limiting procedure in which one eigenvalue is sent to infinity. This parity dependence is a technical feature rather than a change of ensemble, but it is essential in rigorous derivations of partition functions and kernels (Mays, 2012).

3. Global spectral picture and real-eigenvalue statistics

A global law emerges after stereographic projection: in the large-$Y=A^{-1}B$9 limit, the empirical spectral measure of the real spherical ensemble converges to the uniform measure on the sphere. This is the spherical law, directly analogous to the circular law for Ginibre matrices. In the large-$p_{N,M}^{\rm r}$0 limit, the density on the sphere is constant except for a finite contribution on a great circle associated with the real eigenvalues (Mays, 2012).

At finite $p_{N,M}^{\rm r}$1, however, the real ensemble is not fully rotation invariant. Because eigenvalues can be real, the density exhibits reflective symmetry across the real axis, and finite-$p_{N,M}^{\rm r}$2 calculations and simulations show an enhancement along the ring corresponding to the real line after stereographic projection. This corrects a common oversimplification: uniformity on the sphere is an asymptotic statement, not a finite-$p_{N,M}^{\rm r}$3 identity (Mays, 2012).

The real-eigenvalue count is the most studied statistic. If $p_{N,M}^{\rm r}$4 denotes the probability of exactly $p_{N,M}^{\rm r}$5 real eigenvalues, then the expected count satisfies

$p_{N,M}^{\rm r}$6

Thus the typical scale of real eigenvalues is $p_{N,M}^{\rm r}$7, not $p_{N,M}^{\rm r}$8. This sharply contrasts with Hermitian or nearly Hermitian models, in which all eigenvalues are real, and it also distinguishes the real spherical ensemble from the complex spherical ensemble, where the notion of a nontrivial real-eigenvalue sector does not arise (Forrester, 6 Aug 2025).

An explicit density formula for real generalized eigenvalues was also derived in earlier work: $p_{N,M}^{\rm r}$9 where $M$0 is the expected number of real eigenvalues. This expression makes the Cauchy-type weighting along the real line manifest and aligns with the underlying matrix density $M$1 (Mays, 2012).

4. Asymptotic regimes for $M$2

The modern asymptotic theory separates three regimes according to the scale of $M$3: large deviations with $M$4 proportional to $M$5, intermediate deviations with $M$6 proportional to $M$7 but far from the mean, and the local central limit regime with $M$8 in the neighborhood of $M$9 (Forrester, 6 Aug 2025).

Regime	Scaling of $M$0	Leading formulation
Large deviations	$M$1, $M$2	Coulomb gas variational calculus
Intermediate deviations	$M$3, $M$4	Generating function and saddle point
Local CLT	$M$5	Gaussian local central limit theorem

In the large-deviation regime, with $M$6, the leading asymptotic form is

$M$7

The derivation uses a Coulomb gas formalism: $M$8 is related to the minimum electrostatic energy for a charge configuration on a sphere, with charges on the equator representing real eigenvalues and charges in spherical caps representing complex pairs. For small $M$9, the relevant electrostatic energy satisfies

$N$0

so the rate vanishes cubically as $N$1 (Forrester, 6 Aug 2025).

In the intermediate regime, $N$2, the analysis begins from the generating function

$N$3

A contour-integral representation with $N$4, followed by saddle-point analysis, yields

$N$5

where

$N$6

This regime captures deviations on the natural $N$7 scale while remaining outside the local Gaussian window near the mean (Forrester, 6 Aug 2025).

In the local central limit regime, with

$N$8

the distribution of $N$9 is asymptotically Gaussian for $\sqrt N$0. The exponent becomes $\sqrt N$1 precisely when $\sqrt N$2. This local regime was known from earlier work and is incorporated into the recent matching theory as the small-fluctuation endpoint of the intermediate asymptotics (Forrester, 6 Aug 2025).

5. Matching of regimes and the probability of no real eigenvalues

One of the principal structural results is that the three asymptotic regimes are not isolated formulas but match in overlapping windows. On the left tail of the intermediate regime, as $\sqrt N$3,

$\sqrt N$4

the minimizer is

$\sqrt N$5

and the resulting leading asymptotic form is

$\sqrt N$6

If one then sets $\sqrt N$7, this becomes $\sqrt N$8, matching the small-$\sqrt N$9 limit of the large-deviation exponent (Forrester, 6 Aug 2025).

On the right tail of the intermediate regime, as $A B^{-1},$0,

$A B^{-1},$1

and the minimization reproduces the exponential rate in the local central limit theorem. The intermediate regime therefore interpolates smoothly between the $A B^{-1},$2-scale large deviations and the local Gaussian fluctuations near the mean. This suggests that the generating-function saddle point is the correct mesoscopic bridge between electrostatic variational theory and the local CLT (Forrester, 6 Aug 2025).

A particularly important special case is the probability of no real eigenvalues. The leading asymptotic form is

$A B^{-1},$3

It is obtained by setting $A B^{-1},$4 in the generating function and evaluating the resulting integral asymptotically. The appearance of $A B^{-1},$5 matches the leading order for the analogous probability in the real Ginibre ensemble and is conjectured to hold more generally. A common misconception is that the event of having no real eigenvalues should be exponentially small on the $A B^{-1},$6 or $A B^{-1},$7 scale; for this ensemble, the leading decay is instead on the $A B^{-1},$8 scale (Forrester, 6 Aug 2025).

6. Relation to other spherical ensembles and universality

The real Ginibre spherical ensemble is part of a broader family indexed by Dyson’s $A B^{-1},$9. For $A$0, the complex spherical ensemble is determinantal, with points corresponding to the generalized eigenvalues of two appropriately chosen random matrices mapped to the sphere by stereographic projection. Its joint density on $A$1 is

$A$2

and its empirical measure converges almost surely to the uniform measure on the sphere (Alishahi et al., 2014). This complex case provides the fully rotation-invariant benchmark against which the real case is often compared.

For $A$3, the comparison is subtler. Finite-$A$4 spectra are not fully rotation invariant because real eigenvalues occur with nonzero probability and produce a special great circle under stereographic projection. The process is Pfaffian rather than determinantal, and the density shows an enhancement along the real-axis ring. For $A$5, by contrast, there is a depletion along the corresponding ring. In all three cases, however, the large-$A$6 density approaches the spherical law, namely uniformity on the sphere after stereographic projection (Mays, 2012).

Earlier work also emphasizes bulk universality. Away from the real axis and under local scaling, the correlation functions of the real spherical ensemble match those of the real Ginibre ensemble; this is described as the planar limit of the sphere. The geometrical triumvirate of plane, sphere, and anti-sphere therefore organizes several non-Hermitian ensembles into curvature-based universality classes, while preserving the $A$7 hallmark of Pfaffian statistics and mixed real/complex spectra (Mays, 2012).

Taken together, these results identify the real Ginibre spherical ensemble as a model in which global curvature, real-eigenvalue combinatorics, and non-Hermitian universality can all be studied explicitly. Its technical core combines real Schur decomposition, skew-orthogonal polynomials, Pfaffian kernels, generating functions, and Coulomb gas variational methods; its conceptual core is the coexistence of a globally uniform spherical law with a sparse but highly structured real spectrum whose fluctuations are now understood across large, intermediate, and local scales (Forrester, 6 Aug 2025).