The limiting spectral density of the new non-Hermitian beta-ensemble is the logarithm of the radius plus a constant, rotationally invariant on a compact disk.
Thouless formula for random non-Hermitian Jacobi matrices
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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 $\mu$ 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.
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Logarithmic Spectral Distribution of a non-Hermitian $\beta$-Ensemble
The limiting spectral density of the new non-Hermitian beta-ensemble is the logarithm of the radius plus a constant, rotationally invariant on a compact disk.