Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 058, 50 pages      arXiv:2504.12120      https://doi.org/10.3842/SIGMA.2026.058

Logarithmic Spectral Distribution of a Non-Hermitian $\beta$-Ensemble

Gernot Akemann a, Francesco Mezzadri b, Patricia Päßler a and Henry Taylor b
a) Faculty of Physics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Germany
b) School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK

Received May 07, 2025, in final form May 18, 2026; Published online June 18, 2026

Abstract
We introduce a non-Hermitian $\beta$-ensemble and determine its spectral density in the limit of large $\beta$ and large matrix size $n$. The ensemble is given by a general tridiagonal complex random matrix of normal and chi-distributed random variables, extending previous work of Mezzadri and Taylor (2025). The joint distribution of eigenvalues contains a Vandermonde determinant to the power $\beta$ and a residual coupling to the eigenvectors. A tool in the computation of the limiting spectral density is a single characteristic polynomial for centred tridiagonal Jacobi matrices, for which we explicitly determine the coefficients in terms of its matrix elements. In the low temperature limit $\beta\gg1$, our ensemble reduces to such a centred matrix with vanishing diagonal. A general theorem from free probability based on the variance of the coefficients of the characteristic polynomial allows us to obtain the spectral density when additionally taking the large-$n$ limit. It is rotationally invariant on a compact disc, given by the logarithm of the radius plus a constant. The same density is obtained when starting form a tridiagonal complex symmetric ensemble, which thus plays a special role. Extensive numerical simulations confirm our analytical results and put this and the previously studied ensemble in the context of the pseudospectrum. The numerical study of the local nearest-neighbour spacing distribution shows agreement between the tridiagonal ensemble and two-dimensional Poisson statistics (independently of $\beta$), whereas we observe a $\beta$-dependence for the previously introduced ensemble.

Key words: non-Hermitian $\beta$-ensemble; tridiagonal complex random matrix; characteristic polynomial; free probability; logarithmic spectral density.

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