Orthogonal polynomials in a normal matrix model with two insertions
A talk in the Seminar Zufallsmatrizen series by
Sampad Lahiry
| Abstract: | We consider the normal matrix model with external potential $N|z|^2 − 2c N log |z^2+ a^2|$, which represents the Ginibre ensemble with two point charge insertions. As $n, N \rightarrow \infty$ with $n/N \rightarrow t > 0$,the eigenvalues fill out a bounded region in the complex plane, known as the droplet. The average characteristic polynomial satisfies planar orthogonality with respect to a weight supported over the entire complex plane. We show that, for a certain regime of parameters a, c, and t, the limiting zero-counting measure (motherbody) is supported on an interval along the real line, with an asymptotic density characterized by a vector equilibrium problem. We rely on a recent result by Berezin, Kuijlaars, and Parra, which allows us to reformulate the planar orthogonality in terms of non-Hermitian multiple orthogonality (Type I). This opens the door for the steepest descent analysis of the associated Riemann-Hilbert problem. Within the CRC this talk is associated to the project(s): C6 |