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We introduce a dynamical mean-field framework to derive the full finite-time and asymptotic Lyapunov spectra of random recurrent neural networks, accounting for local gain histories and quenched connectivity. This framework yields a closed-contour matrix Dyson equation for the finite-time spectral tail and an operator equation for the stationary cumulative spectrum. Empirical validation via singular-value and QR spectra supports the theoretical predictions, with controlled gain-history replay demonstrating the impact of temporal correlations on instability.
Random RNNs exhibit a rich Lyapunov spectrum beyond the maximal exponent, with finite-time dynamics governed by a matrix Dyson equation and stationary behavior by an operator equation that captures the full history of local gains.
We develop a dynamical mean-field framework for the full finite-time and stationary asymptotic Lyapunov spectra of random recurrent neural networks. At finite observation time, an exact log-determinant representation of tangent-propagator singular values leads to a closed-contour matrix Dyson equation for the smoothed spectral tail. In the stationary setting, a cocycle representation yields an operator equation for the asymptotic cumulative spectrum. Both formulations retain complete local gain histories and the repeated action of quenched connectivity. Direct singular-value calculations and network QR spectra support the predictions in the regimes examined, with sampling uncertainty and numerical cutoffs assessed separately. The stationary equation also recovers the known maximal exponent, and controlled gain-history replay illustrates how temporal correlations redistribute instability. This work was carried out in collaboration with ChatGPT 5.6 Sol. Toward the end of preparation we became aware of David G. Clark's contemporaneous work, \emph{Lyapunov spectrum of random neural networks}[arXiv:2610.12426], which reports overlapping mean-field results for the full Lyapunov spectrum using a cavity approach.