Products of random matrices and generalised quantum point scatterers

Abstract

To every product of $2\times 2$ matrices, there corresponds a one-dimensional Schrödinger equation whose potential consists of generalised point scatterers. Products of {\em random} matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in $\text{SL}(2,\mathbb{R})$. We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.

38 pages, 13 pdf figures. V2 : conclusion added ; Definition of function $\Omega$ changed

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Cited by in corpus (15)

• Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices
• Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices
• Lyapunov exponents for products of complex Gaussian random matrices
• The Lyapunov exponent of products of random $2\times 2$ matrices close to the identity
• Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
• Distributional approach to point interactions in one-dimensional quantum mechanics
• Probability of all eigenvalues real for products of standard Gaussian matrices
• Localization of soft modes at the depinning transition
• Fluctuations of random matrix products and 1D Dirac equation with random mass
• One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers
• Supersymmetric quantum mechanics with Levy disorder in one dimension
• Fluctuations of the product of random matrices and generalized Lyapunov exponent
• Singular behavior of the leading Lyapunov exponent of a product of random $2\times 2$ matrices
• Bounds on Lyapunov exponents via entropy accumulation
• The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results