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Thouless formula for random non-Hermitian Jacobi matrices

Ilya Ya Goldsheid·Boris A Khoruzhenko
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Abstract

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.

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Ilya Ya Goldsheid, Boris A Khoruzhenko. 2003-12-09. Thouless formula for random non-Hermitian Jacobi matrices. https://arxiv.org/abs/math-ph/0312022

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