Perturbative test of single parameter scaling for 1D random media

Abstract

Products of random matrices associated to one-dimensional random media satisfy a central limit theorem assuring convergence to a gaussian centered at the Lyapunov exponent. The hypothesis of single parameter scaling states that its variance is equal to the Lyapunov exponent. We settle discussions about its validity for a wide class of models by proving that, away from anomalies, single parameter scaling holds to lowest order perturbation theory in the disorder strength. However, it is generically violated at higher order. This is explicitely exhibited for the Anderson model.

minor corrections to previous version, to appear in Annales H. Poincare

References in corpus (5)

• Band-center anomaly of the conductance distribution in one-dimensional Anderson localization
• Single parameter scaling in 1-D Anderson localization. Exact analytical solution
• Localization-delocalization transition in a presence of correlated disorder: The random dimer model
• Scattering Theory Approach to Random Schroedinger Operators in One Dimension
• Localization of Phonons in Two Component Superlattice with Random Thicknesses of the Layers

Cited by in corpus (10)

• Absence of extended states in a ladder model of DNA
• The Lyapunov exponent of products of random $2\times 2$ matrices close to the identity
• Upper bounds on wavepacket spreading for random Jacobi matrices
• Scaling diagram for the localization length at a band edge
• Fluctuations of the product of random matrices and generalized Lyapunov exponent
• Scattering Expansion for Localization in One Dimension: from Disordered Wires to Quantum Walks
• Scattering Expansion for Localization in One Dimension
• Low density expansion for Lyapunov exponents
• Universal localization-delocalization transition in chiral-symmetric Floquet drives
• On the equivalence of probability spaces