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

TL;DR

This paper reveals a new mechanism for power law fluctuations in multidimensional systems, showing that non-normal eigenvector amplification causes criticality and scale-free behavior, exemplified in turbulent flow polymer stretching.

Contribution

It introduces a general mechanism based on eigenvector non-normality for criticality in random systems, extending beyond traditional eigenvalue-based explanations.

Findings

Eigenvector amplification increases the effective Lyapunov exponent.

Non-normality lowers the tail exponent, leading to scale-free fluctuations.

The mechanism is demonstrated in turbulent flow polymer stretching.

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.