Planar Equilibrium Measure Problem in the Quadratic Fields with a Point Charge

We consider a two-dimensional equilibrium measure problem under the presence of quadratic potentials with a point charge and derive the explicit shape of the associated droplets. This particularly shows that the topology of the droplets reveals a phase transition: (i) in the post-critical case, the...

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Vydané v:Computational methods and function theory Ročník 24; číslo 2; s. 303 - 332
Hlavný autor: Byun, Sung-Soo
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2024
Predmet:
Analysis
Computational Mathematics and Numerical Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
A non-Hermitian extension of the Marchenko–Pastur law
Primary 30C20
Elliptic Ginibre ensemble
31A25
Planar equilibrium measure problem
Conformal mapping method
Secondary 60B20
Two-dimensional Coulomb gases
Conditional point process
ISSN:1617-9447, 2195-3724
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Popis
Shrnutí:We consider a two-dimensional equilibrium measure problem under the presence of quadratic potentials with a point charge and derive the explicit shape of the associated droplets. This particularly shows that the topology of the droplets reveals a phase transition: (i) in the post-critical case, the droplets are doubly connected domain; (ii) in the critical case, they contain two merging type singular boundary points; (iii) in the pre-critical case, they consist of two disconnected components. From the random matrix theory point of view, our results provide the limiting spectral distribution of the complex and symplectic elliptic Ginibre ensembles conditioned to have zero eigenvalues, which can also be interpreted as a non-Hermitian extension of the Marchenko–Pastur law.
ISSN:1617-9447
2195-3724
DOI:10.1007/s40315-023-00494-4