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arXiv · 2107.07221

Lemniscate ensembles with spectral singularity

Sung-Soo Byun·Seung-Yeop Lee·Meng Yang
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Abstract

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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BibTeXRIS

Sung-Soo Byun, Seung-Yeop Lee, Meng Yang. 2021-07-15. Lemniscate ensembles with spectral singularity. https://arxiv.org/abs/2107.07221

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