Lemniscate ensembles with spectral singularity
Authors:
- Sung-Soo ByunSBSung-Soo Byun
- Seung-Yeop LeeSLSeung-Yeop Lee
- Meng YangMYMeng Yang
Preprint
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 Painlev\'{e} 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.Keywords
All Related Versions
This publication has 46 references indexed in Scilit:
- Progress on the Study of the Ginibre Ensembles2025
- Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point ChargesCommunications on Pure and Applied Mathematics, 2023
- Skew-Orthogonal Polynomials in the Complex Plane and Their Bergman-Like KernelsCommunications in Mathematical Physics, 2021
- A Non-Hermitian Generalisation of the Marchenko–Pastur Distribution: From the Circular Law to Multi-criticalityAnnales Henri Poincare, 2020
- Universality at Weak and Strong Non-Hermiticity Beyond the Elliptic Ginibre EnsembleCommunications in Mathematical Physics, 2018
- Discontinuity in the Asymptotic Behavior of Planar Orthogonal Polynomials Under a Perturbation of the Gaussian WeightCommunications in Mathematical Physics, 2017
- Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the PlaneCommunications on Pure and Applied Mathematics, 2014
- Density of Eigenvalues of Random Normal MatricesCommunications in Mathematical Physics, 2005
- Double scaling limit in the random matrix model: The Riemann‐Hilbert approachCommunications on Pure and Applied Mathematics, 2003
- Regularity of a boundary having a Schwarz functionActa Mathematica, 1991