Eigenvector Geometry as a Universal Amplifier of Heavy-Tailed Fluctuations 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 by the condition number $\kappa$, induces transient growth that increases the effective Lyapunov exponent $\gamma \simeq \gamma_0 + \langle \ln \kappa \rangle$ and lowers the tail exponent $\alpha \simeq -2\gamma / \sigma_{\kappa}^2$, where $\sigma_{\kappa}^2$ is the variance of $\ln \kappa$. 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 two representative systems: (i) polymer stretching in turbulent flows, where intermittent extensions arise from eigenvector amplification of velocity gradients and (ii) financial return distributions, where extending one-dimensional GARCH/Kesten processes to a multidimensional setting yields a collective origin for heavy-tailed market fluctuations and explains their near-universal exponents across assets.