标题： 非厄米特$β$-系综的对数谱分布 显示英文标题

标题： Logarithmic Spectral Distribution of a non-Hermitian $β$-Ensemble

摘要： 我们引入了一个非厄米的$\beta$阵列，并在$\beta$较大和矩阵尺寸$n$较大的情况下确定了其谱密度。该阵列由具有正态分布和卡方分布随机变量的一般三对角复随机矩阵组成，扩展了两位作者之前的工作。特征值的联合分布包含一个幂为$\beta$的范德蒙德行列式和与特征向量的剩余耦合。计算极限谱密度的一个工具是针对中心化的三对角雅可比矩阵的单一特征多项式，我们明确地用矩阵元来确定其系数。在低温极限$\beta\gg1$下，我们的阵列退化为中心化且对角线消失的矩阵。基于特征多项式系数方差的自由概率一般定理使我们能够在进一步考虑大$n$极限时获得谱密度。它是旋转不变的，在一个紧致圆盘上，由半径的对数加上一个常数给出。从一个三对角复对称阵列开始也能得到相同的密度，因此它扮演着特殊角色。广泛的数值模拟证实了我们的分析结果，并将其与先前研究的阵列置于伪谱的背景下。 显示英文摘要

摘要： 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 two of the authors. 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.