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December 2021 Planar orthogonal polynomials and boundary universality in the random normal matrix model
Håkan Hedenmalm, Aron Wennman
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Acta Math. 227(2): 309-406 (December 2021). DOI: 10.4310/ACTA.2021.v227.n2.a3


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.

Funding Statement

H. H. acknowledges support from Vetenskapsrådet (VR) Grant No. 2016-04912.
A. W. acknowledges support from VR Grant No. 2016-04912, Knut and Alice Wallenberg Foundation Grant No. 017.0389, and ERC Advanced Grant No. 692616.


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Håkan Hedenmalm. Aron Wennman. "Planar orthogonal polynomials and boundary universality in the random normal matrix model." Acta Math. 227 (2) 309 - 406, December 2021.


Received: 14 February 2018; Accepted: 5 April 2021; Published: December 2021
First available in Project Euclid: 1 March 2023

Digital Object Identifier: 10.4310/ACTA.2021.v227.n2.a3

Vol.227 • No. 2 • December 2021
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