Sparse Elliptic Random Matrix Models

• Sparse elliptic random matrix models are non-Hermitian matrices with tunable sparsity that interpolate between classical elliptic laws and sparse graph spectra.
• The analysis employs techniques like Hermitization, diagrammatic expansions, and small-ball probability estimates to uncover the limiting spectral distributions and phase transitions.
• These models have wide applications in physics, statistics, and network science, offering insights into universality, eigenvalue localization, and spectral corrections in complex systems.

Sparse elliptic random matrix models constitute a fundamental class of non-Hermitian random matrices characterized by both a prescribed pairwise correlation (interpolating between Hermitian and iid ensembles) and a tunable sparsity parameter. These models interpolate between classical elliptic laws for dense random matrices and the spectral theory of sparse (e.g., Erdős–Rényi) random graphs. They are central to modern random matrix theory due to their universality properties, complex limiting spectral distributions, and wide applicability to problems in physics, statistics, and network science.

1. Model Definition and Structural Features

A sparse elliptic random matrix model is typically defined as follows: for each $n \times n$ matrix $X_n$,

$X_n = E_n \circ M_n,$

where $M_n$ is an elliptic random matrix (off-diagonal entries $(M_{ij}, M_{ji})$ are independent pairs with joint distribution, mean zero, unit variance, and covariance $\mathbb{E}[M_{ij} M_{ji}] = \rho$), $E_n$ is an independent Bernoulli matrix with entries $E_{ij} \sim \mathrm{Bernoulli}(p_n)$ (possibly with $E_{ij} = E_{ji}$ for symmetric sparsity), and $\circ$ denotes the Hadamard (entry-wise) product. The parameter $p_n \in [0,1]$ is the sparsity, so that in the limit $p_n \to 1$ one recovers the classical dense elliptic ensemble, while for $p_n \ll 1$ the matrix is sparse (Carpenter et al., 6 Aug 2025). The standard scaling sets $\mathbb{E}|X_{ij}|^2 \sim 1/(n p_n)$ to maintain nontrivial limiting spectral support as $n \to \infty$.

A key structural property is that the limiting spectral measure and the geometric shape of support (an ellipse or ellipsoid in the complex plane) are functions of both the correlation parameter $\rho$ and the limiting sparsity $p = \lim_{n \to \infty} p_n$.

2. Limiting Spectral Distribution: Sparse Elliptic Law

The central result is the identification of the limiting empirical spectral distribution (ESD) for $X_n$ under appropriate normalization and growth regime $n p_n \to \infty$. Specifically, the ESD (the measure placing $1/n$ at each eigenvalue) converges weakly in probability to the uniform distribution on an ellipsoid: $$\mu_{\text{ell},p,\rho}(dx\,dy) = \frac{1}{\pi a_p b_p} \mathbf{1}_{\left( \frac{x^2}{a_p^2} + \frac{y^2}{b_p^2} \leq 1 \right)} \,dx\,dy,$$ where the axes $a_p$, $b_p$ of the limiting ellipsoid depend not only on the entrywise correlation $\rho$ but also on the asymptotic sparsity $p$ (Carpenter et al., 6 Aug 2025). For $p=1$, this recovers the standard elliptic law, where $a_1 = 1 + \rho$, $b_1 = 1 - \rho$; for $p \to 0$, further technical conditions are needed (often requiring $n p_n \gg \log^C n$ for some $C$) to ensure that connected components in the random graph are sufficiently large such that the ESD is nontrivial.

This result generalizes classical theorems for both the Ginibre (circular) law (Nguyen et al., 2012, Naumov, 2012) and the elliptic law (Nguyen et al., 2012, Alt et al., 2021) by introducing the sparsity parameter as an additional axis of universality. The proof is achieved via Hermitization (expressing the spectral measure in terms of the logarithm of the determinant of a shifted Hermitian matrix built from the singular values), as well as analysis of the least singular value and anti-concentration, in analogy with the methods developed for dense circular and elliptic laws by Girko, Tao–Vu, Rudelson–Tikhomirov, and others (Carpenter et al., 6 Aug 2025, Nguyen et al., 2012, Naumov, 2012).

3. Transition Between Dense and Sparse Regimes

The limiting regime $np_n \to \infty$ is critical for universality. In the dense case ($p_n \to 1$), the standard elliptic law is recovered, and the spectrum fills an ellipse whose axes depend solely on $\rho$. In the genuinely sparse regime ($p_n \ll 1$ but $np_n \to \infty$), the support contracts, and fluctuations in the singular value distribution intensify, but control can be maintained by invoking small-ball probability estimates and using specialized versions of the circular law for sparse matrices (Carpenter et al., 6 Aug 2025, Nguyen et al., 2012, Naumov, 2012).

For extremely sparse settings where $np_n = O(1)$, the spectral distribution may become non-universal, and inhomogeneities, local tree structure, or the percolation threshold (giant component emergence) can result in spectral “outliers” or isolated eigenvalues not captured by the limiting law for $np_n \to \infty$. The interplay of sparsity, entry correlations, and degree distributions is therefore critical.

4. Analytical Techniques and Modifications: Path Integral and Diagrammatic Methods

Beyond the classical resolvent and Hermitization approaches, path integral and diagrammatic expansions provide refined corrections to capture sparse regimes (Baron, 2023). The spectral density $\rho(\lambda)$ in the sparse non-Hermitian case includes corrections in $1/p$ to the standard elliptic law: $$(x^2/a^2) + (y^2/b^2) = 1 - \frac{16}{p} \cdot \frac{\Gamma_4^{(3)}-\Gamma \Gamma_4^{(2)}}{1-\Gamma^2} (x^2/a^2)(y^2/b^2),$$ where the coefficients $\Gamma_4^{(i)}$ encode higher-order moments of the nonzero entries. Here, “ribbon” diagrams, which are subleading in $1/p$, enter the expansion and systematically correct the bulk boundary and density for finite connectivity (Baron, 2023). For products of sparse random matrices or models with non-Gaussian entry distributions, these expansions yield further generalizations, including non-Hermitian analogs of the Marchenko–Pastur law.

Recursion and cavity methods on locally tree-like graphs (Neri et al., 2012) yield closed-form resolvent equations characterizing the local spectral statistics. The support of the spectrum and spectral gap can then be expressed in terms of local graph parameters and symmetry properties of the edge weights.

5. Bulk Universality, Local Laws, and Delocalization

The bulk universality principle holds robustly for sparse elliptic ensembles under appropriate normalization and growth conditions: after rescaling, both the $n$-point correlation statistics and the single eigenvalue gap statistics agree with those of the corresponding dense ensembles (GOE/GUE for Hermitian or Ginibre for non-Hermitian) in local windows of the spectrum (Huang et al., 2015, Alt et al., 2021). Precise local elliptic laws show that the empirical eigenvalue distribution converges to the uniform elliptic measure on scales exceeding the mean eigenvalue spacing (i.e., mesoscopic scales $n^{-\alpha}$ for any $\alpha < 1$), and eigenvectors are completely delocalized—i.e., their mass is spread nearly uniformly over all coordinates (Alt et al., 2021). This delocalization persists as long as $np_n \gg \log^C n$, with possible localization transitions for vanishing $p_n$.

In the presence of heavy-tailed entry distributions, the empirical spectral measure converges to a deterministic measure determined by $\alpha$-stable limits, for $0<\alpha<2$, with the impact of both the tail index and sparsity encoded in the limiting measure (Campbell et al., 2020).

6. Extensions, Applications, and Physical Relevance

Sparse elliptic random matrix models encompass and generalize multiple classical objects, enabling the paper of a variety of phenomena:

• Spectral properties of sparse graphs: When the correlation $\rho = 1$, the model reduces to sparse symmetric (Erdős–Rényi) graphs, while when $\rho=0$, it covers the sparse Ginibre ensemble.
• Vibrational spectra in disordered solids: Block variants capture the Hessian structure of glassy materials (Cicuta et al., 2017).
• Networked dynamical systems: The degree of edge symmetry and distribution of connections impact transport rates and synchronization in complex networks (Neri et al., 2012).
• Random operators and PDEs: Discretizations of random elliptic PDEs via sparse grids or collocation yield sparse random matrices whose spectra encode uncertainty propagation and homogenization (Ernst et al., 2016, Schäfer et al., 2021).

Analytical developments also yield practical algorithms for robust estimation and uncertainty quantification, such as Maronna's M-estimator in elliptical noise, whose spectral properties in high-dimensions are captured by random matrix theory (Kammoun et al., 2014). Sparse grid, multilevel Monte Carlo, and quasi-Monte Carlo methods in high-dimensional stochastic PDEs offer efficient computation of spectral quantities and statistical moments (Giles et al., 2017, Chen et al., 2019).

7. Connections to Universality and Open Problems

Sparse elliptic random matrix models provide an avenue to explore universality at the intersection of correlation structure and sparsity. The full range from dense to sparse uncovers a family of limiting spectral laws parametrized both by correlation and mean degree, with smooth interpolation and, in some cases, critical phase transitions. Open questions remain regarding:

• The precise thresholds and nature of the spectral transition as $np_n$ approaches the connectivity threshold,
• The generalization to models with inhomogeneous degree distributions, nontrivial clustering, or additional correlations (e.g., block structures),
• The emergence of outliers and localization phenomena for extremely sparse or heavy-tailed regimes,
• The development of multidimensional or elliptic deformations of classical ensembles in integrable models and mathematical physics (Mironov et al., 2020, Mironov et al., 2020).

In conclusion, sparse elliptic random matrix models unify and generalize multiple regimes of classical non-Hermitian ensembles, providing a robust framework for understanding the bulk and local spectral properties—including the limiting spectral measure, universality of correlations, and physical manifestations in complex systems (Carpenter et al., 6 Aug 2025, Baron, 2023, Neri et al., 2012, Naumov, 2012, Nguyen et al., 2012, Alt et al., 2021).