Eigenvector Geometry as a New Route to Criticality in Random Multiplicative Systems

Abstract

Heavy-tailed fluctuations and power law distributions pervade physics, biology, and the social sciences, with numerous mechanisms proposed for their emergence. Kesten processes, which are multiplicative stochastic recursions with additive noise or reinjection, provide a canonical explanation, where power law tails arise from transient supercritical excursions as eigenvalues intermittently cross the stability boundary. Here we uncover a distinct and more general mechanism in multidimensional systems: non-normal eigenvector amplification. In random non-normal matrices, the non-orthogonality of eigenvectors, quantified at each time step by the condition number $\kappa_t$ in Kesten-like processes, induces transient growth that increases the effective Lyapunov exponent $\gamma \to \gamma + \mathbb{E}\left[\ln \kappa_t \right]$ and lowers the tail exponent $\alpha \simeq -2\gamma / \sigma_{\kappa}^2$, where $\mathbb{E}\left[\ln \kappa_t \right]$ and $\sigma_{\kappa}^2$ are respectively the mean and variance of $\ln \kappa_t$. As the system dimension $N$ grows, $\kappa$ typically increases proportionally, making non-normal amplification the dominant source of scale-free behavior. We illustrate this mechanism in polymer stretching in turbulent flows, where intermittent extensions arise from eigenvector amplification of velocity gradients.