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This paper analytically computes the Lyapunov spectrum of large random asymmetric recurrent neural networks by reformulating the problem as a response function of tangent-space dynamics, selected via a minimum-norm condition. A cavity method is employed to solve this self-consistently at the $N\to\infty$ limit, revealing that the network's chaos is extensive. This provides access to diffeomorphism-invariant properties of the network dynamics, offering a deeper understanding of their emergent behavior.
Chaos in random neural networks is proven to be extensive, directly linking network size to the proliferation of unstable directions.
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.