Abstract: This research addresses a critical, yet largely unexplored, frontier in the field of non-Hermitian random matrix theory: the regime of strong non-Hermiticity. While the weak non-Hermitian regime is well-documented in existing literature through the works of Fyodorov, Khoruzhenko, and Sommers, the behavior of systems subjected to intense antisymmetric perturbations remains empirically observed but theoretically underserved. In this work, we introduce the Mesoscopic Spectral Projection Theorem. We consider an ensemble of random matrices defined by S = H + gA, where H is a real symmetric matrix and A is a real antisymmetric Gaussian random matrix. By analyzing the limit of strong coupling (g 1), we derive an exact, closed-form analytical expression for the variance of the real parts of the eigenvalues. We rigorously demonstrate that, contrary to expectation, the spectral fluctuations do not diverge; instead, they collapse to a universal deterministic floor, _², which is determined entirely by the spectral invariants (the first two moments) of the unperturbed matrix H. Key Contributions: Analytical Exactness: We provide a formula for the asymptotic spectral variance that remains valid for any finite dimension n, avoiding the need for the thermodynamic limit. Geometric Constraints: The paper establishes that the spectral collapse is a consequence of the rigid algebraic trace constraint of the orthogonal symmetry group O (n), rather than the specific probability distribution of the matrix entries. Universality: We demonstrate that this phenomenon is independent of the initial spectral shape of H and persists across various noise ensembles (Gaussian, Rademacher, and Laplace), confirming the universality of the strong-coupling spectral projection. Research Perspective: By formalizing the "Mesoscopic Crossover" regime (g O (1) ), this paper opens new theoretical avenues for understanding how systems transition from symmetry-governed states to fully localized regimes. This document is submitted to facilitate scientific discussion and serves as a foundational theoretical framework for researchers in mathematical physics, random matrix theory, and complex systems analysis.
Abdelilah AHMOURI (Sun,) studied this question.