Isotropic Brownian motions over complex fields as a solvable model for May–Wigner stability analysis

Ipsen, Jepser R. and Schomerus, Henning Ulrich (2016) Isotropic Brownian motions over complex fields as a solvable model for May–Wigner stability analysis. Journal of Physics A: Mathematical and Theoretical, 49: 385201. ISSN 1751-8113

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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.

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Journal Article
Journal or Publication Title:
Journal of Physics A: Mathematical and Theoretical
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This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Physics A. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at doi:10.1088/1751-8113/49/38/385201
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?? physics and astronomy(all)modelling and simulationmathematical physicsstatistical and nonlinear physicsstatistics and probability ??
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17 Oct 2016 10:32
Last Modified:
18 Dec 2023 01:41