Isotropic Brownian motions over complex fields as a solvable model for May-Wigner stability analysis
| Autor: | Ipsen, J. R., Schomerus, H. |
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| Rok vydání: | 2016 |
| Předmět: | |
| Zdroj: | J. Phys. A 49 (2016) 385201 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1751-8113/49/38/385201 |
| Popis: | We consider matrix-valued stochastic processes known as isotropic Brownian motions, and show that these can be solved exactly over complex fields. While these processes appear in a variety of questions in mathematical physics, our main motivation is their relation to a May-Wigner-like stability analysis, for which we obtain a stability phase diagram. The exact results establish the full joint probability distribution of the finite-time Lyapunov exponents, and may be used as a starting point for a more detailed analysis of the stability-instability phase transition. Our derivations rest on an explicit formulation of a Fokker-Planck equation for the Lyapunov exponents. This formulation happens to coincide with an exactly solvable class of models of the Calgero-Sutherland type, originally encountered for a model of phase-coherent transport. The exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices, and is also closely related to Hermitian matrix models with an external source. Comment: 14 pages |
| Databáze: | arXiv |
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