Transfer operator approach to 1d random band matrices

Abstract: We discuss an application of the transfer operator approach to the analysis of the different spectral characteristics of 1d random band matrices (correlation functions of characteristic polynomials, density of states, spectral correlation functions). We show that when the bandwidth $W$ crosses the threshold $W=N^{1/2}$, the model has a kind of phase transition (crossover), whose nature can be explained by the spectral properties of the transfer operator.