The Random Normal Matrix Model: Insertion of a Point Charge

In this article, we study microscopic properties of a two-dimensional Coulomb gas ensemble near a conical singularity arising from insertion of a point charge in the bulk of the droplet. In the determinantal case, we characterize all rotationally symmetric scaling limits (“Mittag-Leffler fields”) and obtain universality of them when the underlying potential is algebraic. Applications include a central limit theorem for $\log |p_{n}(\zeta )|$ log | p n ( ζ ) | where pn is the characteristic polynomial of an n:th order random normal matrix.