Non-Asymptotic Girko Matrices

• Non-asymptotic Girko matrices are finite-dimensional, non-Hermitian random matrix models with variance-scaled independent entries, permitting explicit finite-N spectral bounds.
• The analysis employs techniques like the zerofreeness method and determinant moment computations to derive O(1) bounds and control eigenvalue outliers.
• The framework bridges classical circular law asymptotics with modern non-asymptotic methods, integrating singular value control and Hermitized linearization.

Searching arXiv for recent and foundational papers on non-asymptotic Girko matrices, Girko-type ensembles, and related non-Hermitian random matrix methods. Non-asymptotic Girko matrices are finite-dimensional non-Hermitian random matrix models studied through explicit probability bounds rather than only large-$n$ limit laws. In the strict sense introduced in “Eigenvalue Bounds for Random Matrices via Zerofreeness” (Mohanty et al., 29 Sep 2025), a non-asymptotic Girko matrix is an $n\times n$ complex random matrix $\mathbf{M}$ with independent off-diagonal entries, zero diagonal, centered entries, and variance bounded by $1/n$. In a broader usage suggested by adjacent literature, the term also covers finite-$N$ analysis of Girko-type ensembles with independent entries, singular-value control for shifted matrices, Hermitized block models, and exact finite-$N$ formulas for non-Hermitian spectra and eigenvectors (Rudelson, 2013). The subject sits at the intersection of classical Girko asymptotics—such as the circular law and Hermitization—and modern non-asymptotic random matrix theory, which asks for dimension-explicit control of spectral radius, singular values, empirical covariance, or joint eigenvalue–eigenvector statistics for matrices of large but fixed size (Vershynin, 2010).

1. Definition and scope

In the narrow definition used in (Mohanty et al., 29 Sep 2025), a non-asymptotic Girko matrix $\mathbf{M}$ has independent complex-valued entries off the diagonal, satisfies

$$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$

and has $\mathbf{M}_{ii}=0$. No further structural assumptions are made: the entries need not be identically distributed, may depend on $n$, and are not assumed subgaussian or to possess bounded higher moments (Mohanty et al., 29 Sep 2025). This definition isolates the variance scale associated with Girko normalization while discarding the asymptotic assumptions that usually underlie the circular law.

The broader literature uses closely related terminology for matrices with independent entries, independent rows, or independent columns under isotropy or subgaussian assumptions, but studied through finite-$n\times n$0 inequalities for singular values and norms rather than through limiting empirical spectral distributions (Rudelson, 2013). In that broader sense, non-asymptotic Girko matrices include the classical i.i.d. square models that under normalization $n\times n$1 satisfy the circular law asymptotically, but whose finite-size behavior is analyzed via smallest singular value estimates, spectral norm bounds, $n\times n$2-net arguments, and Littlewood–Offord theory (Rudelson et al., 2010).

A further extension appears in non-Hermitian linearization frameworks. There, one studies a structured Hermitian block matrix—often called a Girko matrix or Girko linearisation—whose spectrum encodes the singular values or resolvents of a non-Hermitian model. This usage is exact at finite $n\times n$3, but the associated deterministic formulas are often asymptotic or local-law based rather than fully non-asymptotic (Riabov et al., 29 Sep 2025). This suggests that “non-asymptotic Girko matrices” names both a concrete variance-scaled matrix class and a methodological program for finite-dimensional non-Hermitian spectral analysis.

2. Relation to classical Girko theory

Classical Girko theory is asymptotic. For i.i.d. non-Hermitian matrices with the usual $n\times n$4 normalization, the empirical eigenvalue distribution converges to the circular law, and the standard proof architecture studies the logarithmic determinant or the singular values of shifted matrices $n\times n$5. These arguments yield limiting spectral distributions, but they do not by themselves provide quantitative finite-$n\times n$6 guarantees (Rudelson, 2013).

This asymptotic viewpoint is explicit in “New spectral relations between products and powers of isotropic random matrices” (Burda et al., 2012). That paper proves that, in the limit $n\times n$7, the eigenvalue density of the product of $n\times n$8 identically distributed isotropic matrices equals the eigenvalue density of the $n\times n$9-th power of a single matrix from the same ensemble. For products of complex Girko–Ginibre matrices, the limiting planar density is

$\mathbf{M}$0

but the paper emphasizes that this is an asymptotic statement: there are no exact finite-$\mathbf{M}$1 formulas, and finite-size behavior is examined numerically rather than analytically (Burda et al., 2012). The same paper notes that, for finite $\mathbf{M}$2, the spectrum of a product $\mathbf{M}$3 is not equal to that of $\mathbf{M}$4, and that discrepancies are concentrated near the spectral edge.

The non-asymptotic program departs precisely at this point. Instead of deducing only weak convergence of empirical measures, it seeks statements valid for every $\mathbf{M}$5, with absolute constants or explicit failure probabilities. The survey literature formulates this contrast directly: non-asymptotic random matrix theory studies probabilistic bounds for eigenvalues and singular values for random matrices of a large fixed size, whereas asymptotic theory studies limiting spectral behavior as dimensions tend to infinity (Vershynin, 2010). For Girko-type matrices, this shift in viewpoint is especially important because control of finite-$\mathbf{M}$6 smallest singular values of $\mathbf{M}$7 is a core ingredient in circular-law proofs and in quantitative refinements of non-Hermitian spectral analysis (Rudelson et al., 2010).

3. Principal non-asymptotic results

The main theorem presently attached to the strict non-asymptotic Girko class is Theorem 1.2 in (Mohanty et al., 29 Sep 2025). If $\mathbf{M}$8 is a non-asymptotic Girko matrix in the sense above, then its spectral radius satisfies

$\mathbf{M}$9

for some absolute constant $1/n$0 independent of $1/n$1. The same theorem gives an $1/n$2 expected number of outlier eigenvalues: for every $1/n$3,

$1/n$4

where $1/n$5 depends only on $1/n$6, not on $1/n$7 (Mohanty et al., 29 Sep 2025). By Markov’s inequality, these expectation bounds imply constant-probability bounds on the spectral radius and on the number of outliers.

These conclusions are weaker than high-probability asymptotic results for classical i.i.d. Girko ensembles, but they apply under far weaker hypotheses. The paper explicitly contrasts them with refined asymptotic statements such as

$1/n$8

which hold for normalized i.i.d. models but do not extend to arbitrary $1/n$9-dependent, sparse, or highly inhomogeneous entry distributions (Mohanty et al., 29 Sep 2025). The weaker conclusion is not merely technical. The paper discusses examples where circular-law-type behavior fails in this generality, including matrices that are zero with probability $N$0 but have rare huge entries arranged to preserve variance $N$1, and sparse directed Erdős–Rényi-type models whose spectral radius is bounded away from $N$2 with constant probability (Mohanty et al., 29 Sep 2025).

A different non-asymptotic strand concerns singular values rather than eigenvalues. For square i.i.d. subgaussian matrices, the survey literature records the bound

$N$3

while for rectangular $N$4 subgaussian matrices one has

$N$5

(Rudelson et al., 2010). These are not eigenvalue bounds in the non-Hermitian sense, but they are central to Girko’s Hermitization method because singular values of $N$6 control the logarithmic potential and the regularity of the spectrum. The tutorial literature further emphasizes non-asymptotic analogues of the Bai–Yin heuristic,

$N$7

for independent isotropic subgaussian rows or columns (Vershynin, 2010). This suggests that non-asymptotic Girko theory bifurcates into two complementary directions: direct eigenvalue control, exemplified by the zerofreeness method, and singular-value control, which underlies Hermitized analyses.

4. Methods of analysis

The new direct method introduced in (Mohanty et al., 29 Sep 2025) is based on Jensen’s formula and the zerofreeness of the characteristic polynomial. For a matrix $N$8, one considers

$N$9

whose zeros are reciprocals of the nonzero eigenvalues of $N$0. Jensen’s formula and Jensen’s inequality yield, for any matrix $N$1 and any $N$2,

$N$3

Thus spectral radius bounds reduce to controlling an averaged squared determinant on a circle in the complex plane (Mohanty et al., 29 Sep 2025).

For non-asymptotic Girko matrices, the determinant moment can be computed exactly enough because the entries are independent, centered, and variance-bounded. Expanding both determinants in

$N$4

as sums over permutations, independence and zero mean force all cross terms to vanish unless the two permutations coincide. Grouping by the number of non-fixed points then yields

$N$5

uniformly in $N$6 (Mohanty et al., 29 Sep 2025). Plugging this into the Jensen-based inequalities gives the bounded expected spectral radius and bounded expected outlier count.

The singular-value side of non-asymptotic Girko theory uses a different toolkit. The standard ingredients are $N$7-nets, small-ball estimates for linear forms, the decomposition of the sphere into compressible and incompressible vectors, distance-to-random-subspace estimates, and Littlewood–Offord theory via the essential least common denominator $N$8 (Rudelson, 2013). For i.i.d. subgaussian matrices, the distance method reduces lower bounds on $N$9 to estimates on

$\mathbf{M}$0

where $\mathbf{M}$1 is a random normal vector to the span of the first $\mathbf{M}$2 columns. Random normals typically have exponentially large essential LCD, which in turn yields sharp small-ball bounds and the nearly optimal lower tail for the smallest singular value (Rudelson et al., 2010).

A third methodological line uses Hermitian linearization and local laws. In the study of correlated Wishart matrices with overlapping samples, a block matrix

$\mathbf{M}$3

is introduced so that $\mathbf{M}$4 contains the sample covariance matrices on a block diagonal. The associated resolvent $\mathbf{M}$5 is then controlled by matrix Dyson equations and multi-resolvent local laws (Riabov et al., 29 Sep 2025). This is not a non-asymptotic theorem in the zerofreeness sense, but the local laws themselves are finite-$\mathbf{M}$6, high-probability estimates on Hermitized Girko-type matrices. A plausible implication is that non-asymptotic Girko analysis now spans both explicit determinant identities and high-probability resolvent approximations, depending on whether one seeks absolute finite-$\mathbf{M}$7 constants or asymptotically vanishing error terms.

5. Finite-size formulas and structured variants

Exact finite-$\mathbf{M}$8 formulas are uncommon in non-Hermitian random matrix theory, but the Kac–Rice-inspired framework of (Fyodorov, 26 Jun 2025) provides them for several Girko-type ensembles. The central identity rewrites the empirical joint density of an eigenvalue $\mathbf{M}$9 and normalized right eigenvector $$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$0 as

$$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$1

and, after ensemble averaging, expresses the joint probability density $$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$2 through a Hermitized determinant (Fyodorov, 26 Jun 2025). This supplies a finite-$$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$3 alternative to Girko Hermitization for eigenvalue–eigenvector statistics.

For the interpolating Ginibre ensemble

$$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$4

the paper derives an exact finite-$$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$5 joint density $$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$6 and a finite-$$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$7 mean eigenvalue density $$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$8 (Fyodorov, 26 Jun 2025). The formulas interpolate between the standard finite-$$\mathbb{E}\mathbf{M}_{ij}=0,\qquad \mathbb{E}|\mathbf{M}_{ij}^2|\le \frac{1}{n}\qquad (i\ne j),$$9 complex Ginibre density at $\mathbf{M}_{ii}=0$0 and the complex part of the real Ginibre density as $\mathbf{M}_{ii}=0$1. The same framework exposes the emergence of an excess of eigenvalues near the real axis and the weak non-reality scaling regime $\mathbf{M}_{ii}=0$2, in which a limiting density profile near the real line can be written explicitly (Fyodorov, 26 Jun 2025).

For additive deformations $\mathbf{M}_{ii}=0$3, with $\mathbf{M}_{ii}=0$4 complex Ginibre and $\mathbf{M}_{ii}=0$5 deterministic, the same paper proves the exact finite-$\mathbf{M}_{ii}=0$6 formula

$\mathbf{M}_{ii}=0$7

where $\mathbf{M}_{ii}=0$8 and $\mathbf{M}_{ii}=0$9 is an explicit projected quadratic form (Fyodorov, 26 Jun 2025). For rank-one perturbations, the resulting joint density becomes fully explicit in terms of scalar overlaps with the perturbation directions, and the large-$n$0 asymptotics recover outlier formation when $n$1 for $n$2. The paper further derives the typical asymptotic overlap

$n$3

for the outlier eigenvector in the normal rank-one case (Fyodorov, 26 Jun 2025).

These exact finite-$n$4 formulas differ conceptually from the zerofreeness bounds of (Mohanty et al., 29 Sep 2025). The former resolve detailed joint laws for special Gaussian or near-Gaussian ensembles; the latter provides robust $n$5 spectral radius control under minimal second-moment assumptions. Together they illustrate a split within non-asymptotic Girko theory between universal inequalities and integrable finite-size formulas.

6. Extensions, comparisons, and limitations

One important extension concerns non-Hermitian products and ratios. The paper on products and powers of isotropic random matrices is asymptotic, but it clarifies the limiting spectral geometry of Girko–Ginibre products and explicitly notes that finite-$n$6 corrections are concentrated near the spectral edge (Burda et al., 2012). This suggests that any fully non-asymptotic theory for products would need to quantify edge softening rather than merely identify the limiting planar density.

A more recent asymptotic direction studies the ratio $n$7 of two Girko matrices. “On the spectral radius of the ratio of Girko matrices” proves that

$n$8

converges in distribution to a universal heavy-tailed limit, under bounded density, fourth-moment matching, and uniform finite moments (Chafaï et al., 21 Oct 2025). The proof uses Girko Hermitization, local law estimates for Wigner matrices, lower bound estimates on the smallest singular value, and convergence of kernels of determinantal point processes (Chafaï et al., 21 Oct 2025). Although asymptotic rather than non-asymptotic, it reinforces a recurring theme: non-Hermitian radius questions often become tractable only after a careful blend of Hermitization and finite-$n$9 singular-value control.

Another extension arises from intrinsic freeness. For Gaussian matrix sums

$n\times n$00

non-asymptotic spectral norm and spectral inclusion bounds compare $n\times n$01 to a free-probability model $n\times n$02, with error terms governed by variance and covariance parameters such as $n\times n$03, $n\times n$04, and $n\times n$05 (Bandeira et al., 2021). These results do not study Girko matrices directly, but the same paper states that, when combined with a linearization argument, the framework yields strong asymptotic freeness for a remarkably general class of Gaussian random matrix models that may be very sparse, have dependent entries, and lack any special symmetries (Bandeira et al., 2021). A plausible implication is that Hermitian dilations of non-Hermitian Girko-type models could inherit sharp non-asymptotic norm bounds from intrinsic freeness, though that specialization is not carried out explicitly in the cited text.

The main limitations are clear across the literature. The zerofreeness method yields only $n\times n$06 bounds and only constant-probability guarantees without additional structure (Mohanty et al., 29 Sep 2025). Singular-value methods are strongest under subgaussian or finite-moment hypotheses and are often better suited to shifted matrices than to direct eigenvalue counts (Rudelson et al., 2010). Exact finite-$n\times n$07 Kac–Rice formulas currently work most cleanly for Gaussian ensembles and specific deformations (Fyodorov, 26 Jun 2025). Asymptotic product laws and linearized local laws describe large-$n\times n$08 structure elegantly, but they typically do not furnish exact finite-$n\times n$09 formulas or universal constants (Burda et al., 2012). The field therefore remains methodologically plural: no single technique presently covers the full range from worst-case $n\times n$10-dependent entry distributions to eigenvector-resolved finite-size formulas.

In that sense, non-asymptotic Girko matrices are best understood not as a single model class but as a layered research area. At its core lies the strict variance-scaled definition of (Mohanty et al., 29 Sep 2025). Around it sits a larger body of finite-dimensional Girko-type analysis built from smallest singular value estimates, Hermitized block models, exact Gaussian formulas, and determinant-based spectral radius bounds. This suggests a unifying description: non-asymptotic Girko theory studies how much of Girko’s non-Hermitian spectral picture survives when one refuses to pass immediately to the $n\times n$11 limit.