Random matrix products : from the Lyapunov exponent to the generalized Lyapunov exponent
Christophe Texier (LPTMS)
Random matrix products (RMP) appear in various contexts of statistical physics. The most simple property of RMP is the Lyapunov exponent, which measures the typical exponential growth rate.
Focusing on the case of 2*2 matrices, I will provide a brief review of the methods and some known results.
The study of fluctuations of random matrix products can also be relevant in some cases. This requires to introduce the generalized Lyapunov exponent (GLE), i.e. the cumulant generating function of the log of the matrix product. Its determination is much more difficult and requires to identify the largest eigenvalue of a certain non-Hermitian operator.