Acta Mathematica

Volume 227 (2021)

Number 2

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

Pages: 309 – 406



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)


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.

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.

Received 14 February 2018

Received revised 26 June 2020

Accepted 5 April 2021

Published 10 January 2022