Non-Hermitian Ensembles, Circular Law, and Ginibre¶

Job: develop the circular law (Girko Hermitization / Tao–Vu universality), the Ginibre ensembles (complex / real, the DPP kernel, the Edelman real-eigenvalue law), the single ring theorem / the Brown measure, and the elliptic law / non-Hermitian universality — with the circular law via the Hermitization / the free Poisson as the worked example.

Core¶

The circular law (Girko / Tao–Vu)¶

For an $N\times N$ matrix $G$ with i.i.d. entries (mean $0$, variance $1/N$), the empirical spectral distribution $\mu_G=\frac1N\sum\delta_{\lambda_i}$ (the eigenvalues $\lambda_i\in\mathbb{C}$, not real since $G$ is not Hermitian) converges to the circular law — the uniform distribution on the unit disk $\{z:|z|\le1\}$.

Circular law data	Description
The limit	the uniform measure on the unit disk ${z:
Girko Hermitization	reduce to the Hermitian $H_z=(G-z)(G-z)^*$ and the log-determinant
Log-determinant method	$\frac1N\log
Tao–Vu universality	the circular law holds for i.i.d. entries with finite variance (the universality, the same limit for any entry distribution)

Ginibre ensembles¶

The real / complex Ginibre ensemble (the i.i.d. Gaussian entries, the non-Hermitian analog of GOE / GUE). The joint eigenvalue density (the 2D Coulomb gas on $\mathbb{C}$, the Vandermonde repulsion on the complex plane), the determinantal structure (the complex Ginibre is a DPP on $\mathbb{C}$).

Ginibre data	Description
Complex Ginibre	i.i.d. complex Gaussian entries; the joint density $\propto\prod_{i<j}
DPP kernel	$K_N(z,w)=\frac1\pi e^{-(
Limiting density	uniform on the disk ${
Real Ginibre	the eigenvalues are real or come in complex-conjugate pairs; the Edelman $1/\sqrt{N}$ real-eigenvalue proportion

The Hermitization and the logarithmic potential¶

The Girko Hermitization $H_z=(G-z)(G-z)^*$ (the Hermitian matrix associated to $G-z$, the shifted non-Hermitian matrix) and the log-determinant $$ \frac1N\log|\det(G-z)|=\frac1N\log\det H_z^{1/2}=\frac{1}{2N}\mathrm{Tr}\log H_z=\int_0^\infty\log x\,d\mu_{H_z}(x). $$ The circular law is equivalent to the convergence of the log-determinant to $\int_{\mathbb{C}}\log|z-w|\,d\mu(w)=\log|z|$ for $|z|>1$ and $=0$ for $|z|<1$ (the potential theory, the equilibrium measure of the 2D Coulomb gas, the uniform disk). The universality of the circular law (Tao–Vu) via the Hermitization + the local semicircle / Marchenko–Pastur law for $H_z$ + the concentration of the log-determinant.

Hermitization data	Description
$H_z=(G-z)(G-z)^*$	the Hermitization (the shifted Hermitian matrix)
Log-determinant	$\frac1N\log
Limiting $\mu_{H_z}$	the Marchenko–Pastur / the free Poisson with the parameter $
Potential theory	$\int\log x\,d\mathrm{MP}_{

Edge and the single ring theorem¶

The Ginibre edge (the eigenvalues concentrate on $|z|=1$, the boundary of the disk, the edge of the circular law). The single ring theorem (Feinberg–Zee / Guionnet–Krishnapur–Zeitouni): for a general non-Hermitian ensemble $X=U\Sigma V$ (with $U,V$ Haar, $\Sigma$ the singular values), the eigenvalue distribution is a single ring (an annulus) — the Brown measure of the limiting operator. The non-normal / the Brown measure (the Haagerup–Larsen theorem for the Brown measure of the R-diagonal operators, the free probability connection).

Edge / single ring data	Description
Ginibre edge	the eigenvalues concentrate on $
Single ring theorem	$X=U\Sigma V$ (Haar / singular values) → the single ring (the annulus)
Brown measure	the Haagerup–Larsen theorem for the R-diagonal operators (the free probability Brown measure)
R-diagonal	the free probability / the non-normal limiting operators (the free probability connection)

Non-Hermitian universality and the elliptic law¶

The elliptic law (the ensemble interpolating between the Hermitian GOE and the circular Ginibre, the elliptic deformation $G=\frac{1}{\sqrt{N}}((1-\tau)S+\tau A)$ with $S$ symmetric and $A$ antisymmetric, the eigenvalues on the ellipse). The non-Hermitian universality (the elliptic law is universal, the Tao–Vu / the Bourgade–Yau extension).

Elliptic law data	Description
Elliptic ensemble	$G=\frac1{\sqrt N}((1-\tau)S+\tau A)$ (the interpolation Hermitian ↔ circular, $\tau$ the parameter)
The limit	the ellipse ${z:
$\tau=0$	the GOE (the Hermitian / the semicircle on $[-2,2]$)
$\tau=1$	the Ginibre (the circular law on the disk)
Non-Hermitian universality	the elliptic law is universal (the Bourgade–Yau, the same limit for the entry distribution with the same $\tau$)

See it¶

Circular law via the Hermitization and the log-determinant. For the i.i.d. matrix $G$ with variance $1/N$, the eigenvalue density $\mu_G$ converges to the uniform disk. The Girko Hermitization: $\frac1N\log|\det(G-z)|=\int_0^\infty\log x\,d\mu_{H_z}(x)$ (the log-determinant = the integral of $\log x$ against the spectral measure of the Hermitization $H_z=(G-z)(G-z)^*$).

The local semicircle law for $H_z$ (the limiting $\mu_{H_z}$ is the Marchenko–Pastur / the free Poisson with the parameter $|z|^2$ — the $H_z$ is a sample-covariance-type matrix with the parameter $|z|^2$) gives $$ \int_0^\infty\log x\,d\mu_{H_z}\to\int\log x\,d\mathrm{MP}_{|z|^2}=\begin{cases}\log|z|^2&|z|>1\ \text{const}\cdot 0&|z|<1\end{cases} $$ (the potential theory, the free Poisson / the $\square$-transform: $\int\log x\,d\mathrm{MP}_\lambda=\log\lambda$ for $\lambda>1$...). The potential $\frac1N\log|\det(G-z)|\to\log|z|$ for $|z|>1$ and $\to 0$ for $|z|<1$ (the equilibrium measure is uniform on the disk, the logarithmic potential of the uniform disk). This recovers the circular law. The key point: the Hermitization reduces the non-Hermitian circular law to the Hermitian Marchenko–Pastur (via the log-determinant), and the free Poisson / the free probability is the limiting distribution — the free probability is the mechanism behind the circular law, via the Hermitization and the log-determinant.

Check¶

Show that the limiting $\mu_{H_z}$ for the Hermitization $H_z=(G-z)(G-z)^*$ is the Marchenko–Pastur / the free Poisson with the parameter $|z|^2$, and identify the log-determinant computation.

$H_z=(G-z)(G-z)^*=\frac1N(G-z)(G-z)^*$ is the sample-covariance-type matrix of the $N\times N$ matrix $(G-z)/\sqrt N$ (the shifted matrix). For $G$ the i.i.d. matrix with variance $1/N$, the limiting spectral distribution of $H_z$ is the Marchenko–Pastur / the free Poisson with the parameter $\lambda=|z|^2$ (the MP law with the spike $|z|^2$): the free Poisson $\mathrm{MP}_\lambda$ has the density $\rho(x)=\frac{1}{2\pi x}\sqrt{(b-x)(x-a)}$ on $[a,b]$ with $a=(1-\sqrt\lambda)^2$, $b=(1+\sqrt\lambda)^2$ (for $\lambda<1$; with an atom at $0$). The log-determinant $\int\log x\,d\mathrm{MP}_\lambda=\log\lambda$ for $\lambda>0$ (up to the atom at $0$, the $\log 0$ contribution), giving $\int\log x\,d\mu_{H_z}\to\log|z|^2=2\log|z|$ for $|z|>1$ (the potential of the uniform disk is $\log|z|$ for $|z|>1$). The $\frac12$ factor: $\frac1N\log|\det(G-z)|=\frac12\int\log x\,d\mu_{H_z}\to\log|z|$, recovering the circular law potential.

Why is the real Ginibre real-eigenvalue proportion $1/\sqrt N$ (the Edelman law), and what is the structure of the real eigenvalues?

The real Ginibre matrix $G$ (real i.i.d. entries) has eigenvalues that are either real or come in complex-conjugate pairs (the real matrix structure). The expected number of real eigenvalues is $\mathbb{E}[\#\{\text{real eigenvalues}\}]\sim\sqrt{N}$ (the Edelman law), so the proportion of real eigenvalues is $\sim 1/\sqrt N$. The structure: the real eigenvalues are the eigenvalues fixed by the complex conjugation (the real-axis eigenvalues), and the probability that an eigenvalue is real is $\sim 1/\sqrt N$ (the density on the real line vs the density on the disk, the $1/\sqrt N$ ratio). The complex eigenvalues come in pairs $(z,\bar z)$, and the joint density is the 2D Coulomb gas on $\mathbb{C}$ with the real-line constraint (the real Ginibre has a more subtle structure than the complex Ginibre, the Edelman / the Forrester–Nagao computation).

Pitfalls¶

The circular law (Tao–Vu) is for i.i.d. entries with finite variance — the heavy-tail / the elliptic / the dependent cases are separate (the heavy-tail gives the circular law with a different normalization / the Lévy circular law, the elliptic gives the elliptic law, the dependent / the Markov cases are separate).
The Ginibre real-eigenvalue proportion is the Edelman $1/\sqrt N$ (the real Ginibre has a different structure from the complex — the real eigenvalues are a $\sqrt N$-proportion, the complex conjugate pairs the rest).
The single ring theorem is for the $U\Sigma V$ model (the Haar / singular values, the R-diagonal Brown measure), not the general non-normal — the Brown measure of the R-diagonal is the free probability computation, the general non-normal operator needs more.
The elliptic law is the interpolation Hermitian ↔ circular (the $\tau$ parameter, the ellipse); the non-Hermitian universality (Bourgade–Yau) is the same limit for the entry distribution with the same $\tau$, not the same for all $\tau$.
The Girko Hermitization reduces the non-Hermitian to the Hermitian (the $H_z$ and the log-determinant), but the log-determinant concentration (the key step) requires the control of the small singular values (the $\log 0$ / the near-zero eigenvalues of $H_z$), the main technical difficulty (the Tao–Vu polynomial method / the Bourgade–Yau DBM).
The free probability is the limiting mechanism (the asymptotic freeness, the $R$-transform / the $\square$-transform, the Brown measure), but the finite-$N$ fluctuations (the Ginibre edge, the circular law fluctuations) are the RMT (the Ginibre edge universality), not the free probability directly.

Refs¶

Girko, "The circular law: ten years later" (1994); Tao–Vu, "Random matrices: the circular law" (JAMS 2008); "The circular law for i.i.d. entries" (2010)
Ginibre, "Statistical ensembles of complex, quaternion, and real matrices" (J. Math. Phys. 1965); Edelman, "The probability that a random real Gaussian matrix has $k$ real eigenvalues"
Feinberg–Zee, "Non-Hermitian random matrices" (1997); Guionnet–Krishnapur–Zeitouni, "The single ring theorem" (Ann. Math. 2011)
Haagerup–Larsen, "Brown measures of R-diagonal elements in finite von Neumann algebras" (2000); Bourgade–Yau, "The elliptic law" (2017)

Next¶

RMT index · Foundations · Universality & Log-Gases · Determinantal & β-Ensembles · Spiked Models & Statistics · Free probability · FP / subordination — the single ring / the Brown measure via the free harmonic analysis · Functional analysis