Lemniscate ensembles with spectral singularity
arXiv:2107.07221
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.
arXiv commentv1: 29 pages, 5 figures, v2: 35 pages, 4 figures, substantial revision
References in corpus (16)
- Fluctuations of eigenvalues of random normal matrices
- Normal random matrix ensemble as a growth problem
- Duality, Biorthogonal Polynomials and Multi-Matrix Models
- Characteristic Polynomials of Complex Random Matrix Models
- Density of eigenvalues of random normal matrices
- Induced Ginibre ensemble of random matrices and quantum operations
- Topology of quadrature domains
- Concentration for Coulomb gases and Coulomb transport inequalities
- The random normal matrix model: insertion of a point charge
- Universality at weak and strong non-Hermiticity beyond the elliptic Ginibre ensemble
- A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
- A localization theorem for the planar Coulomb gas in an external field
- Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
- The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star
- Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges
- Real eigenvalues of elliptic random matrices