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Lemniscate ensembles with spectral singularity

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 Added by Sung-Soo Byun
 Publication date 2021
  fields Physics
and research's language is English




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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 apply a version of the Christoffel-Darboux identity and the strong asymptotics of the associated orthogonal polynomials, where the latter was obtained by Bertola, Elias Rebelo, and Grava.



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