Fluctuations of the Product of Random Matrices and Generalized Lyapunov Exponent
- PDF / 1,281,363 Bytes
- 62 Pages / 439.37 x 666.142 pts Page_size
- 50 Downloads / 200 Views
Fluctuations of the Product of Random Matrices and Generalized Lyapunov Exponent Christophe Texier1 Received: 20 July 2019 / Accepted: 23 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products Πn = Mn Mn−1 . . . M1 , where Mi ’s are i.i.d. Following Tutubalin (Theor Probab Appl 10(1):15–27, 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 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 ||Πn || and for the variance of ln |ψ(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. Keywords Random matrices · Generalized Lyapunov exponent · Disordered one-dimensional systems · Anderson localisation
1 Introduction The study of random matrix products was initiated by Bellman in 1954 [13] and was later developed by Furstenberg and Kesten [50,51], Guivarc’h and Raugi [60], Le Page [68] and others [84] (see the monograph [22] or the recent one [14]). Among the vast mathematical literature on this topic, one of the key problems is the derivation of sufficient conditions for the central limit theorem to hold. In most of these works, the emphasis is not placed on concrete calculations of the moments of the distribution of the random matrix product. Exception is Ref. [89], where a method for the calculation of the largest Lyapunov exponent
Communicated by Giulio Biroli.
B 1
Christophe Texier [email protected] LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
123
C. Texier
is proposed. Some analytic results can also be obtained by making further assumptions on the matrices. In Ref. [82], Newman has derived the spectrum of Lyapunov exponents for real Ginibre random matrices (strictly speaking, this paper has considered a N -dimensional stochastic linear dynamical system, corresponding to the continuum limit of random matrix products). In the Physics literature, transport properties of disordered waveguides have been studied within transfer matrix approach. Some analytical results were obtained by making some isotropy assumption, i.e. studying phenomenological rather than microscopic models [44,78,79] (see the review [12]). In this spirit, a cla
Data Loading...
