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Lemniscate ensembles with spectral singularity

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arxiv 2107.07221 v2 pith:I7AWY2KV submitted 2021-07-15 math.PR math-phmath.CVmath.MP

classification math.PRmath-phmath.CVmath.MP
keywords lemniscatematrixpointassociatedasymptoticsbertolaboundarycharge
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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 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.

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  1. Free-energy variations for determinantal 2D plasmas with holes

    math-ph 2025-10 conditional novelty 7.0 of 10

    For a determinantal 2D Coulomb gas with small well-separated holes, the correlation energy is independent of hole locations up to O(1), and adding holes changes it by explicit topological log N terms.

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