Local Statistics in Normal Matrix Models with Merging Singularity

We study the normal matrix model, also known as the two-dimensional one-component plasma at a specific temperature, with merging singularity. As the number n of particles tends to infinity we obtain the limiting local correlation kernel at the singularity, which is related to the parametrix of the Painlevé II equation. The two main tools are Riemann–Hilbert problems and the generalized Christoffel–Darboux identity. The correlation kernel exhibits a novel anisotropic scaling behavior, where the corresponding spacing scale of particles is \(n^{-1/3}\) in the direction of merging and \(n^{-1/2}\) in the perpendicular direction. In the vicinity at different distances to the merging singularity we also observe Ginibre bulk and edge statistics, as well as the sine-kernel and the erfc-type kernel (a.k.a. the Faddeeva plasma kernel).