Logarithmic Spectral Distribution of a non-Hermitian β-Ensemble

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 >> 1 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.