Abstract
We extend the method of rescaled Ward identities of Ameur, Kang, and Makarov to study the distribution of eigenvalues close to a bulk singularity, i.e., a point in the interior of the droplet where the density of the classical equilibrium measure vanishes. We prove results to the effect that a certain “dominant part” of the Taylor expansion determines the microscopic properties near a bulk singularity. A description of the distribution is given in terms of the Bergman kernel of a certain Fock-type space of entire functions.
Highlights
We extend the method to allow for a “bulk singularity,” i.e., an isolated point p in the interior of S at which Q = 0
The equilibrium measure σ = σQ is defined as the probability measure that minimizes IQ[μ] over all compactly supported Borel probability measures μ
For k ≥ 2, the Mittag–Leffler potential has a bulk singularity at the origin of type 2k − 2
Summary
In the papers [4,5], the method of rescaled Ward identities was introduced and applied to study microscopic properties of the system {ζ j }n1 close to a (moving) point p ∈ S. The situation in those papers is restricted by the condition that the point p be “regular” in the sense that Q( p) ≥ const. Remark It is well known that the particles {ζ j }n1 can be identified with eigenvalues of random normal matrices with a suitable weighted distribution. The details of this identification are not important for the present investigation. We write D( p, r ) for the open disk with center p and radius r
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
