Logarithmic Spectral Distribution of a non-Hermitian $β$-Ensemble
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| Format: | Preprint |
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2025
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| author | Akemann, Gernot Mezzadri, Francesco Päßler, Patricia Taylor, Henry |
| author_facet | Akemann, Gernot Mezzadri, Francesco Päßler, Patricia Taylor, Henry |
| contents | We introduce a non-Hermitian $β$-ensemble and determine its spectral density in the limit of large $β$ 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 two of the authors. The joint distribution of eigenvalues contains a Vandermonde determinant to the power $β$ 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 $β\gg1$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_12120 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Logarithmic Spectral Distribution of a non-Hermitian $β$-Ensemble Akemann, Gernot Mezzadri, Francesco Päßler, Patricia Taylor, Henry Mathematical Physics Statistical Mechanics Probability We introduce a non-Hermitian $β$-ensemble and determine its spectral density in the limit of large $β$ 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 two of the authors. The joint distribution of eigenvalues contains a Vandermonde determinant to the power $β$ 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 $β\gg1$ 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. |
| title | Logarithmic Spectral Distribution of a non-Hermitian $β$-Ensemble |
| topic | Mathematical Physics Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2504.12120 |