Abstract
A sequence of large invertible matrices given by a small random perturbation around a fixed diagonal and positive matrix induces a random dynamics on a high-dimensional sphere. For a certain class of rotationally invariant random perturbations it is shown that the dynamics approaches the stable fixed points of the unperturbed matrix up to errors even if the strength of the perturbation is large compared to the relative increase of nearby diagonal entries of the unperturbed matrix specifying the local hyperbolicity.
Highlights
A sequence of large invertible matrices given by a small random perturbation around a fixed diagonal and positive matrix induces a random dynamics on a high-dimensional sphere
The paper is thematically located at the interface between random matrix theory, the theory of products of random matrices and random dynamical systems
Let us begin by describing the dynamics (1.1) heuristically
Summary
In a perturbative regime of small coupling of the randomness, one can calculate this localization length [14, 10] and, more generally, the whole Lyapunov spectrum [11, 12] provided the random dynamics of the transfer matrices is well understood. For such systems, one can derive flow equations for the finite volume growth exponents, the so-called DMPK-equations [2, 15, 13]. Dμr,λ(v0) P(vN ∈ B (w)) > 0 , Bξ (u) which proves the claim
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
