The Random Normal Matrix Model: Insertion of a Point Charge
Yacin Ameur, Nam‐Gyu Kang, Seong‐Mi Seo
Abstract
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 $\log |p_{n}(\zeta )|$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>log</mml:mi><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>(</mml:mo><mml:mi>ζ</mml:mi><mml:mo>)</mml:mo><mml:mo>|</mml:mo></mml:math> where p n is the characteristic polynomial of an n :th order random normal matrix.