An infinite transfer matrix approach to the product of random 2 x 2 positive matrices

作者:Bai Zai Qiao*
来源:Journal of Physics A-Mathematical and Theoretical, 2009, 42(1): 015003.
DOI:10.1088/1751-8113/42/1/015003

摘要

This paper concerns the efficient and precise determination of the Laypunov exponent (and other statistical properties) of a product of random 2 x 2 matrices. By considering the ensemble average of an infinite series of regular functions and its iteration, we construct a transfer matrix, which is shown to be a trace class operator in a Hilbert space given that the positiveness of the random matrices is assumed. This fact gives a theoretical explanation of the superior convergence of the cycle expansion of the Lyapunov exponent (Bai 2007 J. Phys. A: Math. Theor. 40 8315). A numerical method based on the infinite transfer matrix is applied to a one-dimensional Ising model with a random field and a generalized Fibonacci sequence. It is found that, in the presence of continuous distribution of a disorder or degenerated random matrix, the transfer matrix approach is more efficient than the cycle expansion method.