Covariance spectra of random linear systems near instability

Abstract

We study how stationary variance is distributed among the principal components of a randomly coupled linear system near instability. The coupling matrix has entries $N^{-1/2}\xi_{ij}$, where the $\xi_{ij}$ are independent copies of a bounded, centered real random variable of variance one. As $N\to\infty$ and the excess damping $\delta_N\to0$, we prove that the covariance spectrum, with eigenvalues rescaled by $\delta_N^2$ and weighted by their shares of the total variance, converges in probability to an explicit beta-product law. The convergence holds throughout $\delta_N\ge N^{-1/2+\varepsilon}$ for every fixed $0<\varepsilon<1/2$. The law describes eigenvalues of order $\delta_N^{-2}$ at ranks of order $N\delta_N^{3/2}$. 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.