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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Abstract
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.