Eigenvalues of large chiral non-Hermitian random matrices

We study a non-Hermitian chiral random matrix of which the eigenvalues are complex random variables. The empirical distributions and the radius of the eigenvalues are investigated. The limit of the empirical distributions is a new probability distribution defined on the complex plane. The graphs of the density functions are plotted; the surfaces formed by the density functions are understood through their convexity and their Gaussian curvatures. The limit of the radius is a Gumbel distribution. The main observation is that the joint density function of the eigenvalues of the chiral ensemble, after a transformation, becomes a rotation-invariant determinantal point process on the complex plane. Then, the eigenvalues are studied by the tools developed by Jiang and Qi [J. Theor. Probab. 30, 326 (2017); 32, 353 (2019)]. Most efforts are devoted to deriving the central limit theorems for distributions defined by the Bessel functions via the method of steepest descent and the estimates of the zero of a non-Trivial equation as the saddle point.