Non-Normal Eigenvector Amplification in Multi-Dimensional Kesten Processes

TL;DR

This work shows that heavy-tailed statistics can arise far from spectral instability via non-normal eigenvector amplification in multidimensional Kesten processes. By deriving relations that link the Lyapunov exponent and tail exponent to eigenvector geometry, specifically the condition number κ, the authors establish γ ≈ γ0 + E[ln κ] and α ≈ -2 γ /(σκ^2) under reasonable independence assumptions. The analysis spans 2D and N-dimensional cases and is corroborated by simulations, with explicit scaling in dimension N and extensions to rotations and reinjection effects. The framework unifies phenomena across turbulence, dynamos, and financial systems, offering a universal mechanism for scale-free behavior without spectral criticality. Collectively, these results define a new universality class for stochastic systems where transient, non-normal amplification drives heavy tails and near-critical statistics in globally stable regimes.

Abstract

Heavy-tailed fluctuations and power law statistics pervade physics, finance, and economics, yet their origin is often ascribed to systems poised near criticality. Here we show that such behavior can emerge far from instability through a universal mechanism of non-normal eigenvector amplification in multidimensional Kesten processes $x_{t+1}=A_t x_t+η_t$, where $A_t$ are random interaction matrices and $η_t$ represents external inputs, capturing the evolving interdependence among $N$ coupled components. Even when each random multiplicative matrix is spectrally stable, non-orthogonal eigenvectors generate transient growth that renormalizes the Lyapunov exponent and lowers the tail exponent, producing stationary power laws without eigenvalues crossing the stability boundary. We derive explicit relations linking the Lyapunov exponent and the tail index to the statistics of the condition number, $γ\!\sim\!γ_0+\lnκ$ and $α\!\sim\!-2γ/σ_κ^2$, confirmed by numerical simulations. This framework offers a unifying geometric perspective that help interpret diverse phenomena, including polymer stretching in turbulence, magnetic field amplification in dynamos, volatility clustering and wealth inequality in financial systems. Non-normal interactions provide a collective route to scale-free behavior in globally stable systems, defining a new universality class where multiplicative feedback and transient amplification generate critical-like statistics without spectral criticality.

Figures (4)

• Figure 1: Simulation of the two-dimensional Kesten process \ref{['eq:apx_kesten']}, with matrices $\mathbf{A}_t$ defined by \ref{['eq:apx_kesten_2d_num']}. Parameters are set to $\delta = 0$ and $\ln\rho = -1$ (hence $\sigma_\rho = 0$). Left column: no rotation ($\theta_t = 0$). Middle column: uniform rotation ($\theta_t \sim \mathcal{U}(0,2\pi)$). Each row corresponds to a different level of non-normal variability, from top to bottom: $\sigma_z = 0.5, 1, 1.5$. The $y$-axis reports the crossing mean $\langle x\rangle$ between the two components of the system. The right panel shows the complementary cumulative distribution function (CCDF) of the absolute mean, normalized so that the fifth-largest observation equals 1. Solid lines correspond to the no-rotation case, while dashed lines correspond to uniform rotations. The numerical procedures used to estimate the Lyapunov and tail exponents are detailed in Appendix \ref{['apx:tools']}.
• Figure 2: Study of the two-dimensional Kesten process \ref{['eq:apx_kesten']}, with matrices $\mathbf{A}_t$ defined by \ref{['eq:apx_kesten_2d_num']}. The system is simulated with $\ln\rho_t=-1$ (top) and $\ln\rho_t \sim \mathcal{N}(-1,1/9)$ (bottom), while varying the non-normal variance $\sigma_z$ and the imbalance parameter $\delta$, both with uniform rotations ($+$) and without rotations ($\cdot$). Theoretical predictions with rotations (dashed lines) and without rotations (solid lines) are taken from \ref{['eq:apx_case_2']} and \ref{['eq:apx_case_1']}, respectively. We considered the case without any imbalance i.e. $\delta=0$; (black) and with imbalance i.e. $\delta=0.1$; (blue). Left panel: Lyapunov exponent $\gamma$ as a function of $\sigma_z$. Right panel: corresponding tail exponent $\alpha$ (see Section \ref{['apx:tools']} for details of the numerical estimation). We zoomed the results on $\alpha\in[0,7]$, where the missing points are due to a diverging measure of the tail exponent, when it tends to be close to zeros and/or the Lyapunov exponent becomes positive.
• Figure 3: Lyapunov exponent $\gamma$ for the Kesten process \ref{['eq:apx_kesten']} with matrices $\mathbf{A}_t$ generated via the decomposition \ref{['eq:apx_dec_n_dim']} as a function of $\sigma \sqrt{\ln N}$, testing the scaling predictions. For the top panel, we generated $\mathbf{V}_t^\dag \Lambda_t \mathbf{V}_t$ once at the beginning and kept it fixed throughout the simulation, assuring that the "normal" part of the matrix is fixed. In the bottom panel we used \ref{['eq:apx_dec_n_dim']} and generated the matrix $\mathbf{V}_t^\dag\boldsymbol{\Lambda}_t\mathbf{V}_t$ at each step. We set $\ln\lambda_{i,t}\overset{\text{i.i.d.}}{\sim}\mathcal{N}(-1,1/9)$ and $\ln s_{i,t}\overset{\text{i.i.d.}}{\sim}\mathcal{N}(0,\sigma^2)$. Each curve shows $\gamma$ as a function of $\sigma$ for different dimensions $N$. The "No rotation" fixes $\mathbf{U}_t=\mathbf{I}$; the "Uniform rotation" samples $\mathbf{U}_t$ uniformly at each time. Results are obtained using the QR-based Lyapunov estimator described in Appendix \ref{['apx:tools']}.
• Figure 4: Lyapunov exponent $\gamma$ for the Kesten process \ref{['eq:apx_kesten']} with matrices $\mathbf{A}_t$ generated via the decomposition \ref{['eq:apx_dec_n_dim']}, as a function of the system dimension $N$. For the top panel, we generated $\mathbf{V}_t^\dag \Lambda_t \mathbf{V}_t$ once at the beginning and kept it fixed throughout the simulation, ensuring that the "normal" part of the matrix remained constant. In the bottom panel, we used \ref{['eq:apx_dec_n_dim']} and generated the matrix $\mathbf{V}_t^\dag\boldsymbol{\Lambda}_t\mathbf{V}_t$ at each step. We set $\ln\lambda_{i,t}\overset{\text{i.i.d.}}{\sim}\mathcal{N}(-1,1/9)$ and $\ln s_{i,t}\overset{\text{i.i.d.}}{\sim}\mathcal{N}(0,\sigma^2)$. Each curve shows $\gamma$ as a function of $N$ for different values of $\sigma$. The "No rotation" fixes $\mathbf{U}_t=\mathbf{I}$; the "Uniform rotation" samples $\mathbf{U}_t$ uniformly at each time. Results are obtained using the QR-based Lyapunov estimator described in Appendix \ref{['apx:tools']}.