Girko's Elliptic Law in Random Matrix Theory

• Girko’s Elliptic Law is a large‑n theorem that establishes the limiting elliptical distribution of eigenvalues for non‐Hermitian random matrices with correlated off-diagonals.
• The law demonstrates universality as the ellipse’s axes depend solely on the first four moments and the correlation parameter ρ, bridging the circular and Hermitian regimes.
• Its proof framework utilizes Hermitization and specialized control over small singular values to ensure convergence, even on mesoscopic scales.

Girko’s Elliptic Law is the large-$n$ limit theorem for the empirical spectral distribution of non-Hermitian random matrices whose off-diagonal symmetric positions are allowed to be correlated. In its standard form, if $X_n$ is scaled as $n^{-1/2}X_n$, then the empirical spectral distribution of its eigenvalues converges weakly in probability to the uniform probability measure on an ellipse in the complex plane; the axes of the ellipse are determined by the correlation parameter $\rho=\mathbb E(X_{12}X_{21})$, and the limiting law does not depend on the specific distribution of the entries, in this sense it is universal (Naumov, 2012). In the complex setting, the empirical spectral distribution of $\frac{1}{\sqrt n}X_n$ converges almost surely to the uniform probability measure on an ellipse or ellipsoid, giving a generalization of the circular law for ensembles interpolating between i.i.d. non-Hermitian and Hermitian symmetry (Nguyen et al., 2012).

1. Canonical random-matrix formulation

For the real ensemble treated in “Elliptic law for real random matrices” (Naumov, 2012), the matrix $X_n=(X_{ij})_{1\le i,j\le n}$ satisfies the following structural assumptions. The pairs $(X_{ij},X_{ji})$ for $i\neq j$ are i.i.d. random vectors; $\mathbb E X_{12}=\mathbb E X_{21}=0$; $\mathbb E X_{12}^2=\mathbb E X_{21}^2=1$; $X_n$0; $X_n$1 with $X_n$2; and the diagonal entries $X_n$3 are i.i.d., independent of the off-diagonal, with mean zero and finite variance.

The empirical spectral distribution is defined by

$X_n$4

where $X_n$5 are the eigenvalues of $X_n$6 and $X_n$7 is Borel. The theorem states that $X_n$8 converges weakly in probability to a deterministic probability measure $X_n$9 supported on an ellipse (Naumov, 2012).

For the complex formulation in “The Elliptic Law” (Nguyen et al., 2012), the entries above the diagonal are i.i.d. copies of a pair of complex random variables $n^{-1/2}X_n$0, the diagonal entries are i.i.d. with zero mean and finite variance, and the pairs $n^{-1/2}X_n$1 are independent across $n^{-1/2}X_n$2. The relevant moment conditions are

$n^{-1/2}X_n$3

Under these general assumptions, the empirical spectral distribution of $n^{-1/2}X_n$4 converges almost surely to the elliptic law (Nguyen et al., 2012).

2. Limiting measure, geometry, and special cases

For real $n^{-1/2}X_n$5, the limiting support is the ellipse

$n^{-1/2}X_n$6

and the limiting measure is uniform on this ellipse with density

$n^{-1/2}X_n$7

(Naumov, 2012, Nguyen et al., 2012).

The ellipse has major axis $n^{-1/2}X_n$8 and minor axis $n^{-1/2}X_n$9 (Naumov, 2012). The parameter $\rho=\mathbb E(X_{12}X_{21})$0 is the correlation between $\rho=\mathbb E(X_{12}X_{21})$1 and $\rho=\mathbb E(X_{12}X_{21})$2, so the geometry of the support is determined by the correlation between transposed positions.

Two special cases organize much of the surrounding theory. If $\rho=\mathbb E(X_{12}X_{21})$3, the ellipse becomes the unit circle, and the elliptic law reduces to the circular law. If $\rho=\mathbb E(X_{12}X_{21})$4, the ellipse degenerates to a line segment, corresponding to the Wigner semi-circular law for symmetric matrices (Naumov, 2012). In this sense, the elliptic law interpolates between the circular law and the Hermitian regime (Nguyen et al., 2012).

For complex $\rho=\mathbb E(X_{12}X_{21})$5 with $\rho=\mathbb E(X_{12}X_{21})$6, the ellipsoid is rotated appropriately. The rotated support is given by

$\rho=\mathbb E(X_{12}X_{21})$7

(Nguyen et al., 2012).

3. Universality and the status of rigorous proofs

A central feature of Girko’s Elliptic Law is universality. In the real theorem, the limiting law does not depend on the specific distribution of the entries as long as the assumptions above are satisfied; it depends on the first four moments and the correlation $\rho=\mathbb E(X_{12}X_{21})$8, not on higher moments or finer details of the distribution (Naumov, 2012). The paper explicitly emphasizes that neither higher moments nor absolute continuity are needed, and that the law applies to both continuous and discrete entries, including Gaussian and Bernoulli examples (Naumov, 2012).

“The Elliptic Law” by Hoi H. Nguyen and Sean O’Rourke gives the first full proof of the elliptic law under very general and minimal restrictions, extending the state of rigor achieved for the circular law to the correlated-entry setting (Nguyen et al., 2012). Their formulation covers a broad class of random complex matrices and establishes almost sure convergence of the empirical spectral distribution to the uniform probability measure on an ellipse or ellipsoid (Nguyen et al., 2012).

This rigorous development also clarifies the relation to Girko’s earlier program. Girko initiated the study of empirical spectral distributions for ensembles beyond Hermitian and i.i.d. non-Hermitian models, and formulated both the circular law and the elliptic law. The later rigorous treatments isolate the technically decisive point: control of small singular values and uniform integrability of logarithmic singular-value statistics (Nguyen et al., 2012).

4. Hermitization, logarithmic potential, and singular values

The standard proof strategy is Girko’s Hermitization. Rather than analyzing eigenvalues of a non-Hermitian matrix directly, one studies the singular values of the shifted matrix $\rho=\mathbb E(X_{12}X_{21})$9 and reconstructs the eigenvalue distribution through logarithmic potentials (Naumov, 2012).

For the empirical measure $\frac{1}{\sqrt n}X_n$0, the logarithmic potential is

$\frac{1}{\sqrt n}X_n$1

The proof shows convergence of $\frac{1}{\sqrt n}X_n$2 to the logarithmic potential of the uniform measure on the ellipse (Naumov, 2012).

The singular-value problem is the technical core. For the real theorem, control over least singular values is established using small-ball probability estimates and careful decompositions of the unit sphere into compressible and incompressible vectors (Naumov, 2012). In the broader complex proof, lower bounds on the least singular value are obtained using inverse Littlewood-Offord theory, generalized arithmetic progression structure, truncation, and comparison with Gaussian ensembles (Nguyen et al., 2012). These steps are used to prove uniform integrability of the logarithm and to transfer explicit Gaussian calculations to general entry distributions.

The methodology therefore has two coupled components. One component is potential theory, via Hermitization and the logarithmic potential. The other is a least-singular-value analysis strong enough to control the unstable part of the logarithm. The resulting framework is the basic mechanism behind both the global elliptic law and later local refinements (Naumov, 2012, Nguyen et al., 2012).

5. Local elliptic law and eigenvector delocalization

The global law concerns fixed macroscopic sets in $\frac{1}{\sqrt n}X_n$3. “Local elliptic law” strengthens this by proving convergence on mesoscopic scales slightly above the typical eigenvalue spacing in the bulk spectrum, with an optimal convergence rate (Alt et al., 2021).

In that setting, $\frac{1}{\sqrt n}X_n$4 is an $\frac{1}{\sqrt n}X_n$5 elliptic random matrix with complex entries such that for $\frac{1}{\sqrt n}X_n$6, $\frac{1}{\sqrt n}X_n$7 are i.i.d. copies of a pair $\frac{1}{\sqrt n}X_n$8 with zero mean, unit variance, and $\frac{1}{\sqrt n}X_n$9, where $X_n=(X_{ij})_{1\le i,j\le n}$0 controls the correlation between $X_n=(X_{ij})_{1\le i,j\le n}$1 and $X_n=(X_{ij})_{1\le i,j\le n}$2 (Alt et al., 2021). The limiting density is still

$X_n=(X_{ij})_{1\le i,j\le n}$3

with

$X_n=(X_{ij})_{1\le i,j\le n}$4

(Alt et al., 2021).

The local statement is formulated for mesoscopic observables

$X_n=(X_{ij})_{1\le i,j\le n}$5

with $X_n=(X_{ij})_{1\le i,j\le n}$6. For $X_n=(X_{ij})_{1\le i,j\le n}$7 in the bulk ellipse and $X_n=(X_{ij})_{1\le i,j\le n}$8, the deviation between the empirical sum and the deterministic integral is bounded with very high probability by

$X_n=(X_{ij})_{1\le i,j\le n}$9

which is optimal up to the $(X_{ij},X_{ji})$0 factor (Alt et al., 2021). The law therefore holds down to scales slightly above the typical eigenvalue spacing in the bulk.

A corollary is complete delocalisation for the corresponding eigenvectors in any basis. If $(X_{ij},X_{ji})$1 denotes the set of eigenvectors with eigenvalues in the bulk ellipse, then for any deterministic vector $(X_{ij},X_{ji})$2,

$(X_{ij},X_{ji})$3

uniformly for $(X_{ij},X_{ji})$4, with high probability (Alt et al., 2021). In the terminology of that paper, the overlaps match what would be expected for a uniformly random unit vector.

6. Extensions, neighboring laws, and structural contrasts

Several later results place Girko’s Elliptic Law inside a wider non-Hermitian theory. For products of $(X_{ij},X_{ji})$5 independent real random matrices with correlated off-diagonal pairs, “On one generalization of the elliptic law for random matrices” proves that the limit distribution of the empirical spectral distribution does not depend on $(X_{ij},X_{ji})$6 and equals the distribution of the $(X_{ij},X_{ji})$7th power of the random variable uniformly distributed on the unit disc (Götze et al., 2014). This contrasts sharply with the single-matrix case $(X_{ij},X_{ji})$8, where $(X_{ij},X_{ji})$9 determines the support and density through the ellipse.

Sparse non-Hermitian random matrices exhibit a different kind of extension. “Spectra of sparse non-Hermitian random matrices: an analytical solution” gives an exact analytical expression for the spectrum of a sparse, partially-oriented regular graph ensemble and describes it as a sparse realization of Girko’s elliptic law (Neri et al., 2012). In the dense limit, after rescaling $i\neq j$0 and taking $i\neq j$1, the formula yields a uniform elliptic support matching Girko’s law for appropriate identification of parameters. At finite sparsity, however, the eigenvalue density is non-uniform inside the ellipse, the support and density profile depend on the sparsity parameter $i\neq j$2 and on the degree of orientational asymmetry, and isolated eigenvalues may appear outside the continuous bulk (Neri et al., 2012).

A useful contrast is provided by the local inhomogeneous circular law. There the entries are independent and centered, with a general variance profile $i\neq j$3, and the limiting density is typically inhomogeneous, determined by a self-consistent system rather than by a uniform elliptic measure. That work explicitly does not treat correlations; it generalizes Girko’s circular law to matrices with an arbitrary flat inhomogeneous variance profile, not to elliptic cases (Alt et al., 2016). This distinction addresses a common misconception: ellipticity is not merely non-uniform planar support, but specifically the correlated-entry deformation leading to a uniform measure on an ellipse.

More recent work also develops analogs of Girko-type laws for level crossings of random matrix pencils $i\neq j$4. In the Hermitian setting, the elliptic law enters through the induced elliptic parameter $i\neq j$5, and the resulting limiting level-crossing density is not uniform but is expressed through the logarithmic energy of the elliptic law (Shapiro, 28 Apr 2026). This shows that the elliptic law now functions not only as a global eigenvalue law for a single non-Hermitian matrix, but also as an input to potential-theoretic formulas in broader spectral-degeneracy problems.