The Thouless formula for random non-Hermitian Jacobi matrices

Random non-Hermitian Jacobi matricesJ _n of increasing dimensionn are considered. We prove that the normalized eigenvalue counting measure ofJ _n converges weakly to a limiting measure μ asn→∞. We also extend to the non-Hermitian case the Thouless formula relating μ and the Lyapunov exponent of the second-order difference equation associated with the sequenceJ _n . The measure μ is shown to be log-Hölder 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.