Lemniscate ensembles with spectral singularity

Abstract

We consider a family of random normal matrix models whose eigenvalues tend to occupy lemniscate type droplets as the size of the matrix increases. Under the insertion of a point charge, we derive the scaling limit at the singular boundary point, which is expressed in terms of the solution to the model Painlevé IV Riemann-Hilbert problem. For this, we introduce a version of the Christoffel-Darboux identity and combine it with the strong asymptotics of the associated orthogonal polynomials due to Bertola, Elias Rebelo and Grava.

v1: 29 pages, 5 figures, v2: 35 pages, 4 figures, substantial revision

References in corpus (2)

• Normal random matrix ensemble as a growth problem
• Real eigenvalues of elliptic random matrices

Cited by in corpus (3)

• Universal scaling limits of the symplectic elliptic Ginibre ensemble
• Wronskian structures of planar symplectic ensembles
• Scaling Limits of Planar Symplectic Ensembles