Random 2D linear cocycles II: statistical properties

2025/04/30 by Duarte, Pedro, Durães, Marcelo, Graxinha, Tomé +1
#37A30 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2505.00146

Abstract

Consider the space of two dimensional random linear cocycles over a shift in finitely many symbols, with at least one singular and one invertible matrix. We provide an explicit formula for the unique stationary measure associated to such cocycles and establish a Furstenberg-type formula characterizing the Lyapunov exponent. Using the spectral properties of the corresponding Markov operator and a parameter elimination argument, we prove that Lebesgue almost every cocycle in this space satisfies large deviations estimates and a central limit theorem.

Citations

• A dynamical Thouless formula
• Schrödinger operators with dynamically defined potentials
• Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles
• Fine properties of the integrated density of states and a quantitative\n separation property of the Dirichlet eigenvalues
• Holder Continuity of the Integrated Density of States for Quasi-Periodic Schrodinger Equations and Averages of Shifts of Subharmonic Functions
• Random matrix products and measures on projective spaces

Related

• Random 2D linear cocycles I: dichotomic behavior
• Large deviation type estimates for random cocycles
• Lyapunov Exponents of Linear Cocycles Over Markov Shifts
• A random cocycle with non Hölder Lyapunov exponent
• Continuity of Lyapunov Exponents for Random 2D Matrices