Fine asymptotic behavior for eigenvalues of random normal matrices: ellipse case (Q2795577)

The author gives a detailed study of the random normal matrices with quadratic external potentials where the associated orthogonal polynomials are Hermite polynomials and the limiting support (called droplet) of the eigenvalues is an ellipse. Considerations are given for a set of \(n\) point particles, \({z_1, \dots , z_n}\), in the complex plane \(\mathbb{C}\) that interact via two-dimensional (i.e., logarithmic) Coulomb repulsion and are subjected to a quadratic confining potential \(V(z)\). They can form a special case of random process called Coulomb gas or \(\beta\)-ensemble. The density of these particles is supported approximately on an elliptical shape converging, for \(n\rightarrow\infty\), to the characteristic function. By Theorem 1.1, the authors are able to present a detailed behavior of the density for the potential \(V(z)\). This theorem has many applications: the study of the number of particles outside the ellipse (``escaping'' particles), the average number of the escaping particles that scales as \(\sqrt{n}\), the number of escaping particles ``per unit arclength'' of the ellipse.NEWLINENEWLINEThe whole paper is divided into five sections. After some preliminaries in Section I, there is a presentation of the list of the notations and a few useful facts including the asymptotic behavior of the orthogonal polynomial \(p_n(z)\) using the properties of the Hermite polynomials. The next section gives a proof of Theorem 1.1. Then, in Section IV, a proof of another Theorem 1.2 is provided. The last section is devoted for the discussion about important issues including some corrections in Theorem 1.1. There are also four appendices.