Covariance spectra of random linear systems near instability

We study how stationary variance is distributed among the principal components of a randomly coupled linear system near instability. The coupling matrix has entries N-1/2ξᵢⱼ, where the ξᵢⱼ are independent copies of a bounded, centered real random variable of variance one. As N→∞ and the excess damping δN→0, we prove that the covariance spectrum, with eigenvalues rescaled by δN² and weighted by their shares of the total variance, converges in probability to an explicit beta-product law. The convergence holds throughout δN≥ N-1/2+ε for every fixed 0<ε<1/2. The law describes eigenvalues of order δN⁻² at ranks of order NδN3/2. Its limiting rank profile decays as r-4/3 at large normalized rank and determines how many principal components retain a prescribed fraction of the stationary variance. The proof combines an exact computation of planar moments, uniform estimates for growing matrix words that accommodate nonsymmetric entry distributions, and resolvent bounds controlling the contribution of long times.