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 $κ$, induces transient growth that increases the effective Lyapunov exponent $γ\simeq γ_0 + \langle \ln κ\rangle$ and lowers the tail exponent $α\simeq -2γ/ σ_κ^2$, where $σ_κ^2$ is the variance of $\ln κ$. As the system dimension $N$ grows, $κ$ 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.

Figures (1)

• Figure 1: Effect of stochastic non-normality on system critical behavior. (a) Schematic illustration of non-normal instability in a two-dimensional system of the form \ref{['eq:matrix_2d_0']}, with constant spectral radius $\rho$. When the non-normal parameter switches periodically as $z_t \to 1/z_t$, the dominant mode alternates between the two axes, leading to cumulative reinjections and instability. (b) Empirical Lyapunov and tail exponents measured for a two-dimensional Kesten process \ref{['eq:matrix_2d']}, with constant spectral radius $\ln \rho = -1$ and non-normal asymmetry $\ln z_t \sim \mathcal{N}(0,\sigma_z^2)$, so that $\mathbb{E}[\ln \kappa] = \sqrt{2/\pi}\,\sigma_z$. Symbols ($\cdot$) correspond to the case without rotations ($\mathbf{U}(\theta_t)=\mathbf{I}$), and ($+$) to uniform random rotations in the plane. Black solid and dashed lines show the theoretical predictions without and with rotations, respectively. The red horizontal line marks the critical threshold $\gamma=0$. (c) Same as (b) but generalized to $N$ dimensions (\ref{['eq:apx_dec_n_dim23']}) for $N=6,\,10,\,20$. The $x$-axis corresponds to the expected log-condition number $\mathbb{E}[\ln \kappa] \sim \sigma \sqrt{\ln N}$ in the extreme-value limit, with i.i.d. log-normal singular values $\ln s_{i,t} \sim \mathcal{N}(0,\sigma^2)$. The numerical procedures used to estimate Lyapunov and tail exponents are detailed in the SM.