The Random Normal Matrix Model: Insertion of a Point Charge

Abstract

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.

Scaling Limits of Planar Symplectic Ensembles

“…In the generic case of a potential of Mittag-Leffler type we will be able to provide an explicit expression for the limiting origin kernel, that differs from the Ginibre universality class. Our findings can be thought of as the counterparts for previous results in random normal matrix ensembles [15,25]. (See also [16,18,42] for extensive studies on the orthogonal polynomials associated with Mittag-Leffler type potentials.…”

“…An important feature of this equation is that in the large-N limit, it does not depend on the choice of potential Q if p is regular, in that sense that 0 < ∆Q(p) < ∞ is non-vanishing and bounded. Due to this property, Ward's equation has been utilised to show the local universality conjectures in various situations, see, e.g., [10,14,15] and references therein. To be more precise, the overall strategy for the universality proof using Ward's equation is as follows.…”

“…Derivation of Ward's equation (cf. [13, Theorem 1.3], [14, Lemma 2], [15,Lemma 3.1]). The first step is to show that for a C 2 -smooth potential Q and a regular point p, the following form of Ward's equation holds:…”

“…Structure of the correlation kernel (cf. [13, Theorem 1.1], [15,Theorem 1.3]). The next step is to show that for a general potential Q and a regular point p, the limiting correlation kernel K exists and is of the form…”

“…Remark 2.11 (limiting Ward's equation at a singular point of Mittag-Leffler type). As an analogue of [15], we briefly discuss the limiting form of Ward's equation at a singular point of Mittag-Leffler type. For λ > 0, let Q be a potential satisfying…”