Additively Deformed GOE Matrix

• Additively deformed GOE matrices are real symmetric random matrices perturbed by deterministic or low-rank structures, altering their typical Gaussian behavior.
• These models exhibit phase transitions in their extreme eigenvalue statistics, including outlier formation and novel large deviation regimes.
• Advanced techniques like variational principles, resolvent analysis, and replica methods offer quantitative insights with broad applications in inference, optimization, and physical systems.

An additively deformed GOE matrix is a real symmetric random matrix whose law is altered by adding a deterministic or randomly structured matrix, typically of fixed or low rank, to the classical Gaussian Orthogonal Ensemble (GOE). Variants also arise by deforming specific parameters, such as diagonal variances. These models are central in high-dimensional statistics, spin glass theory, signal processing, theoretical ecology, and mathematical physics. They serve both as testbeds for extreme-value and localization phenomena, and as quantitative models for optimization landscapes, statistical inference, and physical networks.

1. Formal Definition and Variants

Let $G$ denote an %%%%1%%%% GOE matrix: it is real symmetric, with entries $G_{ij} \sim \mathcal{N}(0, 1)$ for $i < j$, and $G_{ii} \sim \mathcal{N}(0, 2)$. An additively deformed GOE matrix takes the form

$X = G + D$

where $D$ is a deterministic (or in some models, random but structured) symmetric matrix, often diagonal or of low rank. The deformation can also involve modifying the diagonal variances (e.g., in (Jiang et al., 19 Oct 2024), the “deformation” means changing $G_{ii} \sim \mathcal{N}(0,\sigma)$ with $\sigma \ge 0$ instead of the GOE value $2$).

Prominent examples include:

• Rank-$r$ spike model: $D = VDV^*$ where $V$ is $N \times r$ with orthonormal columns, $D$ is $r \times r$ diagonal; see (Knowles et al., 2012).
• Principal minor deformation: The analysis of largest eigenvalues in all minor submatrices of $X$ with deformed diagonal variance (Jiang et al., 19 Oct 2024).
• Ecological (Lotka–Volterra) models: Interaction matrix $\Sigma = \kappa W + (\alpha/n)11^\top$ where $W$ is GOE, $11^\top$ the rank–one mean interaction (Gueddari et al., 17 Oct 2025).
• GOE–GUE crossover models: Deformation interpolates between real symmetric (GOE) and Hermitian (GUE), e.g., $H = S + i\alpha A$ (Kanazawa, 2020).
• Quadratic optimization and statistical detection: $X = G + \text{diag}(v_1, ..., v_N)$ with possible outlier, leading to BBP transition phenomena (Boursier et al., 17 Aug 2024, Peng, 2012, Doussal, 15 Mar 2025).

The spectral distribution of $X$ always converges, in the large $N$ limit and under mild regularity, to the semicircular law in the bulk. The new phenomena introduced by the deformation are overwhelmingly manifested in the behavior of the largest, smallest, or outlier eigenvalues and associated eigenvectors.

2. Spectral Properties and Extreme Value Statistics

Extreme Eigenvalue Statistics

For deformations $D$ of fixed rank and bounded norm, the largest (or smallest) eigenvalues of $X$ undergo phase transitions and exhibit nontrivial large deviation behavior. Classical results show that for $D$ sufficiently large, an “outlier” eigenvalue separates from the semicircle support: $\theta(d) = d + \frac{1}{d}$ for spike $d$ satisfying $d - 1 \gg N^{-1/3}$, as shown in (Knowles et al., 2012). The fluctuation of the rescaled outlier is then asymptotically Gaussian, with variance scaling as $N^{-1/2}(d-1)^{1/2}$. Overlapping outliers (multiple spikes with nearby values) induce joint eigenvalue statistics governed by an explicit finite-dimensional random matrix model with covariance proportional to $(d+1)/d^2$ (GOE case).

For deformations via diagonal variance (as in (Jiang et al., 19 Oct 2024)):

• The largest eigenvalue of 2×2 principal minors, taken over all minors, has a limiting distribution that is Gumbel for $\sigma \in [0,2]$ and, when $\sigma > 2$, a new explicitly given law (see formula (2.3) in (Jiang et al., 19 Oct 2024)).
• Normalized, the statistic

$$A_{p, \sigma} \left( \max_{i<j} \ell_{i, j} - B_{p, \sigma} \right) \xrightarrow{w} G_\sigma(z)$$

with $G_\sigma$ the limiting cumulative distribution, and $A_{p, \sigma}, B_{p, \sigma}$ normalization constants depending on the dimension and variance.

This “phase transition” at $\sigma = 2$ reflects a marked change in the tail behavior of the extreme eigenvalue distribution.

Large Deviations

Rigorous rate functions $I_{\nu,t}^\Lambda(\lambda)$ for smallest eigenvalues (and similarly for largest) are formulated in (Boursier et al., 17 Aug 2024) and (Doussal, 15 Mar 2025), utilizing free probability (subordination functions $\omega_{\nu, t}(z)$ and functionals of the log potential $S_\nu(x)$): $$I_{\nu,t}^{\Lambda}(\lambda) = \frac{1}{2}[S_\nu(\omega_{\nu, t}(\lambda)) - S_\nu(\gamma(\Lambda, \lambda))] + \frac{1}{4t} [ (\lambda - \gamma(\Lambda, \lambda))^2 - (\lambda - \omega_{\nu, t}(\lambda))^2 ]$$ for $\lambda$ up to the left edge of the free convolution, with $\gamma(\Lambda, \lambda)$ encoding whether the outlier escapes the bulk (BBP phase).

For largest eigenvalues and full-rank diagonal perturbations, the replica method gives the rate function as a Legendre transform of the logarithmic Laplace/Cumulant generating function (Doussal, 15 Mar 2025),

$$P(\lambda) \asymp \exp[-\beta N L(\lambda)], \quad L(\lambda) = \max_{s \ge 0} [s\lambda - \phi(s)]$$

where $\phi(s)$ is computed by the spherical partition function in the replica symmetric regime.

3. Methods of Analysis

Several advanced probabilistic and analytic techniques are deployed:

• Min–Max/Variational characterization: Eigenvalues are extremal values of Rayleigh quotients over high-dimensional spheres; the analysis thus draws upon concentration of measure for Gaussian processes (Peng, 2012).
• Stein–Poisson approximation and Hájek projection: Used in extreme-value statistics for principal minors (Jiang et al., 19 Oct 2024) and beyond. Conditioning and U-statistics are linearized via Hájek projection, facilitating tractable bounds and limit theorems.
• Green function/resolvent analysis: The isotropic local semicircle law controls fluctuations of resolvent entries, essential to major results in outlier distribution (Knowles et al., 2012).
• Supersymmetry and saddle-point integration: For averages of characteristic polynomials and resolvent statistics, a blend of Grassmann integration and saddle-point methods is central, particularly for physical applications (Fyodorov et al., 2014).
• Replica and Parisi variational frameworks: In spin glass and ecological modeling, replica symmetric and Ruelle probability cascade approaches yield the free energy and characterize the hierarchical/gibbsian structure of invariant distributions (Doussal, 15 Mar 2025, Gueddari et al., 17 Oct 2025).
• Concentration inequalities and eigenvector statistics: Extreme value theory for quadratic forms of eigenvectors is reduced to Gaussian computations under Haar-orthogonality (Erdos et al., 2022).

4. Applications in High-Dimensional Statistics and Signal Processing

Deformed GOE models arise naturally in numerous statistical problems:

• Wishart approximation: Extreme eigenvalue statistics for principal minors of a high-dimensional Wishart matrix can be linked via a central limit theorem to those of a suitably deformed GOE (Jiang et al., 19 Oct 2024).
• Signal detection/spiked covariance: The BBP (Baik–Ben Arous–Peche) phase transition in principal component analysis is mathematically captured by the outlier behavior in spiked deformed GOE matrices (Knowles et al., 2012, Boursier et al., 17 Aug 2024).
• Random optimization landscapes: Quadratic optimization with random fields involves the largest eigenvalues of deformed matrices, with rate functions indicative of the rare event distributions (Doussal, 15 Mar 2025).
• Quantum chaotic systems: Properties of deformed GOE matrices, specifically matrix element distributions and averages of half-integer powers of characteristic polynomials, determine scattering statistics (Fyodorov et al., 2014) and describe transitions (GOE–GUE crossover; (Kanazawa, 2020)).

5. Interplay with Physical and Ecological Models

Deformed GOE matrices function as realistic models for random interactions in complex physical and ecological systems:

• Spin glass landscapes and random Gibbs measures: In theoretical ecology, the interaction matrix in generalized Lotka–Volterra SDEs is naturally modeled as a deformed GOE. The unique invariant distribution is a random Gibbs measure whose free energy, computed via spin glass techniques, reflects marginal stability and glassy energy landscapes (Gueddari et al., 17 Oct 2025).
• Hamiltonian structure and stability: The quadratic form defined by the deformed GOE interaction matrix shapes the ecosystem’s energy landscape: transitions between unique equilibrium and marginally stable configurations are governed by the regime of interaction parameters (mean field and fluctuations).
• Stochastic dynamics and universality: Random Schrödinger operators in condensed matter and Anderson models on antitrees and thin boxes in $\mathbb{Z}^3$ employ deformed Laplacians, yielding GOE statistics in eigenvalue processes (Sadel, 2017, Sadel et al., 2014).

6. Open Questions and Directions for Future Research

Notable open problems include:

• Principal minor statistics for higher dimensions: For $k$-by-$k$ principal minors with $k \ge 3$, extension of the limiting law (Gumbel for $\sigma \le 2$, new regime for $\sigma > 2$) beyond $k = 2$ remains unresolved; cubic and quartic roots are algebraically feasible, but the behavior for $k > 4$ is open (Jiang et al., 19 Oct 2024).
• Threshold effects and universality in minor statistics: Is there a universal $k_0$ so that for $k \ge k_0$, the limiting distribution always reverts to Gumbel irrespective of diagonal variance? The general answer awaits further development.
• Extensions to non–Hermitian and non–Gaussian ensembles: Though many results hold for real symmetric matrices, the adaptation to complex, non-Hermitian, or heavy-tailed settings is an active area.
• Marginal stability and glassy phases in Gibbsian ecology: Bridging the full phase diagram between unique equilibrium and marginally stable states—correspondence with Parisi ultrametric structure, geometric transitions in free energy functional order parameters—remains to be detailed.
• Statistical inference and random matrices: The direct implications for hypothesis testing, detection, estimation, and optimization in high-dimensional inference warrant continued investigation, especially in the presence of strong outlier eigenvalues or deformed backgrounds.

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In summary, the paper of additively deformed GOE matrices at the intersection of random matrix theory, probability, statistical physics, and applied mathematics has produced a rich set of results on spectral statistics, phase transitions, universality, and limiting distributions. The techniques developed—ranging from variational principles to advanced probabilistic approximations—are adaptable to a broad class of problems in theory and application, with ongoing research exploring further ramifications in higher dimensions, model generality, and practical signal processing.