The Thouless formula for random non-Hermitian Jacobi matrices

Random non-Hermitian Jacobi matrices J(n) of increasing dimension n are considered. We prove that the normalized eigenvalue counting measure of J(n) converges weakly to a limiting measure mu as n -> infinity. We also extend to the non-Hermitian case the Thouless formula relating mu and the Lyapunov exponent of the second-order difference equation associated with the sequence J(n). The measure mu is shown to be log-Holder continuous. Our proofs make use of (i) the theory of products of random matrices in the form first offered by H. Furstenberg and H. Kesten in 1960 [8], and (ii) some potential theory arguments.