The random normal matrix model: insertion of a point charge

Abstract

In this article, we study microscopic properties of a two-dimensional eigenvalue ensemble near a conical singularity arising from insertion of a point charge in the bulk of the support of eigenvalues. In particular, we characterize all rotationally symmetric scaling limits ('Mittag-Leffler fields') and obtain universality of them when the underlying potential is algebraic. Applications include a result on the asymptotic distribution of $\log |p_n(\zeta )|$ where $p_n$ is the characteristic polynomial of an $n$:th order random normal matrix.

In this version, we have expanded the range of possible singularities, so that we in particular include the full, two-parametric family of Mittag-Leffler fields. Potential Anal (2021)

References in corpus (5)

• Characteristic polynomials of complex random matrices and Painlevé transcendents
• Maximum of the characteristic polynomial of the Ginibre ensemble
• The high temperature crossover for general 2D Coulomb gases
• Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
• Two-Dimensional Elliptic Determinantal Point Processes and Related Systems

Cited by in corpus (17)

• A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
• The high temperature crossover for general 2D Coulomb gases
• Universal scaling limits of the symplectic elliptic Ginibre ensemble
• Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials
• Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
• Universality of the number variance in rotational invariant two-dimensional Coulomb gases
• Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges
• Two-Dimensional Elliptic Determinantal Point Processes and Related Systems
• Exponential moments for disk counting statistics of random normal matrices in the critical regime
• Scaling Limits of Planar Symplectic Ensembles
• Wronskian structures of planar symplectic ensembles
• Spherical Induced Ensembles with Symplectic Symmetry
• Partial Isometries, Duality, and Determinantal Point Processes
• Lemniscate ensembles with spectral singularity
• Functional Central Limit Theorems for Constrained Mittag-Leffler Ensemble in Hard Edge Scaling
• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• Orthogonal polynomials in the normal matrix model with two insertions