Eigenvector Geometry as a New Route to Criticality in Random Multiplicative Systems
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866915800776966144 |
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| author | Troude, Virgile Sornette, Didier |
| author_facet | Troude, Virgile Sornette, Didier |
| contents | 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 $κ_t$ in Kesten-like processes, induces transient growth that increases the effective Lyapunov exponent $γ\to γ+ \mathbb{E}\left[\ln κ_t \right]$ and lowers the tail exponent $α\simeq -2γ/ σ_κ^2$, where $\mathbb{E}\left[\ln κ_t \right]$ and $σ_κ^2$ are respectively the mean and variance of $\ln κ_t$. 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 polymer stretching in turbulent flows, where intermittent extensions arise from eigenvector amplification of velocity gradients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21755 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenvector Geometry as a New Route to Criticality in Random Multiplicative Systems Troude, Virgile Sornette, Didier Chaotic Dynamics Data Analysis, Statistics and Probability 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 $κ_t$ in Kesten-like processes, induces transient growth that increases the effective Lyapunov exponent $γ\to γ+ \mathbb{E}\left[\ln κ_t \right]$ and lowers the tail exponent $α\simeq -2γ/ σ_κ^2$, where $\mathbb{E}\left[\ln κ_t \right]$ and $σ_κ^2$ are respectively the mean and variance of $\ln κ_t$. 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 polymer stretching in turbulent flows, where intermittent extensions arise from eigenvector amplification of velocity gradients. |
| title | Eigenvector Geometry as a New Route to Criticality in Random Multiplicative Systems |
| topic | Chaotic Dynamics Data Analysis, Statistics and Probability |
| url | https://arxiv.org/abs/2510.21755 |