2016/02/29 by J. R. Ipsen, J R Ipsen, H. Schomerus +1 · 16 citations
Computer Science · Mathematics · Physics and Astronomy · #Biorthogonal system #Brownian motion #Eigenvalues and eigenvectors #Hermitian matrix #Isotropy #Lyapunov function #Phase (matter) #Quantum Information and Cryptography #Random Matrices and Applications #Stability (learning theory) #Statistical Mechanics and Entropy #Type (biology) #cond-mat.dis-nn #cond-mat.stat-mech #math-ph #math.MP
paper · pdf · doi:10.1088/1751-8113/49/38/385201
published in Journal of Physics A Mathematical and Theoretical 49(38), 385201 (Institute of Physics) · 14 pages
openalex created_date 2016/06/24 · openalex publication_date 2016/08/30 · arxiv created 2017/08/22 · arxiv updated 2017/08/23 · openalex updated_date 2026/08/05
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.