Planar orthogonal polynomials and boundary universality in the random normal matrix model

Håkan Hedenmalm Dept. of Mathematics, Royal Institute of Technology, Stockholm, Sweden; Dept. of Mathematics & Computer Sciences, St. Petersburg State University, St. Petersburg, Russia; and Dept. of Mathematics & Statistics, University of Reading, United Kingdom Aron Wennman Department of Mathematics, Stockholm University, Stockholm, Sweden

TBD mathscidoc:2203.43011

Acta Mathematica, 227, (2), 309-406, 2021.12
We obtain an asymptotic expansion of planar orthogonal polynomials with respect to exponentially varying weights, relevant for random matrix theory. As a consequence we show that the density of states in the random normal matrix model has universal error function transition across smooth interfaces for the limiting eigenvalue density. The key ingredient in the proof of the asymptotic expansion is the construction of the orthogonal foliation flow: a smooth flow of closed curves foliating a neighborhood of the interface, where the weighted L^2-mass of the orthogonal polynomial concentrates. The defining property of the flow is that the orthogonal polynomial of a given degree with respect to the disintegrated weighted area measure on the curves is approximately stationary in the flow parameter. To compute the coefficient functions in the expansion, we develop an algorithm which collapses the orthogonality relations to conditions along the interface, which become inhomogeneous Toeplitz kernel equations.
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@inproceedings{håkan2021planar,
  title={Planar orthogonal polynomials and boundary universality in the random normal matrix model},
  author={Håkan Hedenmalm, and Aron Wennman},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310114736882845928},
  booktitle={Acta Mathematica},
  volume={227},
  number={2},
  pages={309-406},
  year={2021},
}
Håkan Hedenmalm, and Aron Wennman. Planar orthogonal polynomials and boundary universality in the random normal matrix model. 2021. Vol. 227. In Acta Mathematica. pp.309-406. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220310114736882845928.
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