Isotropic Brownian motions over complex fields as a solvable model for MayWigner stability analysis
Abstract
We consider matrixvalued 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 MayWignerlike stability analysis, for which we obtain a stability phase diagram. The exact results establish the full joint probability distribution of the finitetime Lyapunov exponents, and may be used as a starting point for a more detailed analysis of the stabilityinstability phase transition. Our derivations rest on an explicit formulation of a FokkerPlanck equation for the Lyapunov exponents. This formulation happens to coincide with an exactly solvable class of models of the CalgeroSutherland type, originally encountered for a model of phasecoherent 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 September 2016
 DOI:
 10.1088/17518113/49/38/385201
 arXiv:
 arXiv:1602.06364
 Bibcode:
 2016JPhA...49L5201I
 Keywords:

 Mathematical Physics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics
 EPrint:
 14 pages