Thouless formula for random non-Hermitian Jacobi matrices

Goldsheid, Ilya Ya; Khoruzhenko, Boris A

Thouless formula for random non-Hermitian Jacobi matrices

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Description

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 $μ$ as $n\to\infty$. 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 sequence $J_n$. The measure $μ$ is shown to be log-Hölder continuous.
14 pages