Lyapunov spectrum of random neural networks

Abstract

The Lyapunov spectrum of a nonlinear recurrent neural network with random asymmetric couplings is calculated in the limit $N\to\infty$. The calculation is based on a finite-$N$ identity that expresses the cumulative distribution of Lyapunov exponents as a response function of the tangent-space dynamics, with the tangent trajectory selected by a minimum-norm condition rather than by an initial condition. A cavity method then determines this response function at large $N$ through a self-consistent single-site problem. This result establishes that the chaos in this network is extensive and gives access to diffeomorphism-invariant properties of the dynamics. This work was done in collaboration with the AI models GPT-6 Astra and Claude Opus 5.5.