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 Π_{n}=M_{n}M_{n 1}…M_{1} , where M_{i}'s are i.i.d. Following Tutubalin (Theor Probab Appl 10(1):1527, 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 SL (2 ,R ) where explicit calculations are possible. For concreteness, I study in detail transfer matrix products for the onedimensional 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 Π_{n} and for the variance of lnψ (x ) , where ψ (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.
 Publication:

Journal of Statistical Physics
 Pub Date:
 August 2020
 DOI:
 10.1007/s1095502002617w
 arXiv:
 arXiv:1907.08512
 Bibcode:
 2020JSP...181..990T
 Keywords:

 Random matrices;
 Generalized Lyapunov exponent;
 Disordered onedimensional systems;
 Anderson localisation;
 Mathematical Physics;
 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 LaTeX, 71 pages, 10 pdf figures