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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Abstract	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.
AbstractList	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.
Author	Byun, Sung-Soo
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Cites_doi	10.1007/s00365-022-09566-5 10.1142/S0217751X96000432 10.1016/j.jat.2014.04.002 10.1112/plms/pds032 10.1214/13-AAP980 10.1103/PhysRevD.64.114021 10.1111/sapm.12537 10.1007/s00220-012-1508-x 10.1088/1361-6544/aca3f4 10.1007/s00365-015-9283-5 10.1088/1751-8121/ab1af9 10.1007/BF02188664 10.1090/jams828 10.1007/s00365-020-09517-y 10.1088/0305-4470/35/31/308 10.1016/j.aim.2022.108600 10.1007/s00220-021-04230-8 10.1063/1.4939973 10.1007/s00365-003-0536-3 10.1088/0951-7715/28/2/347 10.4171/rmi/1340 10.1016/j.aim.2019.04.010 10.1007/BF01013377 10.1002/cpa.21541 10.1142/S2010326311500092 10.1007/s10955-019-02276-6 10.1515/9781400835416 10.1063/1.1704292 10.1007/s00220-017-2888-8 10.1016/j.aim.2012.03.021 10.1007/s00220-017-3027-2 10.1007/s00365-016-9356-0 10.1007/BF02398888 10.1007/s00023-020-00973-7 10.1016/j.aim.2015.06.020 10.1063/5.0089789 10.3842/SIGMA.2018.091 10.1090/pcms/026/08 10.1007/978-3-319-72456-0_10 10.1137/22M1531361 10.1007/s00220-023-04673-1 10.3842/SIGMA.2023.033 10.3842/SIGMA.2022.007 10.1007/s00208-023-02603-z 10.4171/JST/474 10.3842/SIGMA.2023.020 10.1007/978-3-662-03329-6 10.1016/B978-0-08-010067-8.50007-4 10.1063/5.0086835 10.1142/S2010326322500472
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Snippet	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...
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SubjectTerms	Analysis Computational Mathematics and Numerical Analysis Functions of a Complex Variable Mathematics Mathematics and Statistics
Title	Planar Equilibrium Measure Problem in the Quadratic Fields with a Point Charge
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