Exact Finite-n Real-Part Variance Floor and a Pseudospectral 1/g Correction in the Real Elliptic Ginibre Ensemble

We study the statistics of the real parts of the eigenvalues of the real elliptic Ginibre ensemble, written as S = H + gA, where H is a GOE-type symmetric matrix, A is an antisymmetric Gaussian matrix, and g ≥ 0 is a non-Hermiticity coupling.

We prove from first principles that the variance of the real parts possesses an exact finite-n floor as g → ∞:

    σ²∞ = (n−2) / [2(n−1)],

obtained by Gaussian (Wick) contraction of quadratic forms together with an exact trace identity, and verified by three independent numerical routes.

We further show that the approach to this floor is NOT the symmetric Lorentzian law σ²∞ + (Var_H − σ²∞)/(1+g²) one might posit, but contains a genuine 1/g term, established here at the 8σ level (an even-only expansion is rejected). We trace this term to the non-normality of S: the typical eigenvector overlap obeys the Chalker–Mehlig scaling O_kk − 1 ≈ n/g², while the real-part variance is controlled by the heavy tail of the overlap distribution, whose exponent is the critical value α = 1 (consistent with the Bourgade–Dubach theorem). We identify the carriers of this tail as the persistently-defective real eigenvalues of S, whose summed squared real parts decay exactly as 1/g.

As context we include an elementary "No-Vacuum" lemma (S + Sᵀ = I ⟹ Re λ = 1/2), shown to deliver set-symmetry about Re = 1/2 but not to pin individual eigenvalues. Throughout, we separate what is proved, what is measured with error bars, and what remains open.

A complete reproducibility script (Python) regenerating every number and figure from a fixed random seed is available at: [https://github.com/PHOTON-COURIER-NABIL]

Author: Ahmouri Abdelilah (ORCID: 0009-0005-4398-7655), Independent Researcher, Issoire, France.

The numerical experiments, symbolic checks, and figures were produced with the assistance of an AI system (Claude, Anthropic) used as a computational instrument under a strict no-fitting, no-hidden-correction discipline; all research questions, interpretation, and the separation of proved from conjectured statements are the responsibility of the author.