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 $n\to\infty$. 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-H\"older continuous.