Fluctuations of the product of random matrices and generalized Lyapunov exponent

Abstract

I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products ${\Pi }_n=M_n{M}_{n-1}\cdots M_1$, where $M_i$'s are i.i.d.. Following Tutubalin [Theor. Probab. Appl. {\bf 10}, 15 (1965)], the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group. The formalism is illustrated by considering products of random matrices from the group $\mathrm{SL}(2,\mathbb{R})$ where explicit calculations are possible. For concreteness, I study in detail transfer matrix products for the one-dimensional Schrödinger equation where the random potential is a Lévy noise (derivative of a Lévy process). In this case, I obtain a general formula for the variance of $\ln ||{\Pi }_n||$ and for the variance of $\ln |\psi (x)|$, where $\psi (x)$ is the wavefunction, in terms of a single integral involving the Fourier transform of the invariant density of the matrix product. Finally I discuss the continuum limit of random matrix products (matrices close to the identity ). In particular, I investigate a simple case where the spectral problem providing the generalized Lyapunov exponent can be solved exactly.

LaTeX, 71 pages, 10 pdf figures ; v3: few corrections, references added, conclusion slightly extended

References in corpus (16)

• Anderson Transitions
• Band-center anomaly of the conductance distribution in one-dimensional Anderson localization
• Functionals of the Brownian motion, localization and metric graphs
• Scaling and the center of band anomaly in a one-dimensional Anderson model with diagonal disorder
• Universal distribution of Lyapunov exponents for products of Ginibre matrices
• The Lyapunov exponent of products of random $2\times 2$ matrices close to the identity
• Anomalies and non-log-normal tails in one-dimensional localization with power-law disorder
• Products of random matrices and generalised quantum point scatterers
• Asymptotics of finite system Lyapunov exponents for some random matrix ensembles
• Perturbative test of single parameter scaling for 1D random media
• Non-monotonic disorder-induced enhanced tunneling
• Localization for one-dimensional random potentials with large local fluctuations
• Breaking supersymmetry in a one-dimensional random Hamiltonian
• Statistics of quantum transmission in one dimension with broad disorder
• Generalized Lyapunov exponent of random matrices and universality classes for SPS in 1D Anderson localisation
• Cycle-expansion method for the Lyapunov exponent, susceptibility, and higher moments

Cited by in corpus (9)

• Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
• Dyson's disordered linear chain from a random matrix theory viewpoint
• Anomalous scaling of heterogeneous elastic lines: a new picture from sample to sample fluctuations
• Transmission of waves through a pinned elastic medium
• The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results
• Equilibrium properties of two-species reactive lattice gases on random catalytic chains
• Representation theory and products of random matrices in $\text{SL}(2,\mathbb{R})$
• Evolution of grammatical forms: some quantitative approaches
• Disordered harmonic chain with random masses and springs: a combinatorial approach