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On the roots of generalised Hermite polynomials and their distribution

Pieter Roffelsen
Wednesday, November 6, 2019 - 11:30 to 12:30

The generalized Hermite polynomials are a countable family of polynomials with applications to random matrices, quantum mechanics and nonlinear wave equations. Central in each of these applications is the fact that these polynomials generate rational solutions of the nonlinear ODE which is the fourth of the celebrated six Painlev\'e equations. About fifteen years ago, Peter Clarkson published numerical investigations of their roots and posed the problem of describing their distribution in the complex plane. In this talk, I will discuss an asymptotic approach to this problem yielding an explicit, uniform, asymptotic description of the bulk of the roots in terms of elliptic functions as the degree of the corresponding polynomial grows large. The roots, after proper rescaling, densely fill up a bounded quadrilateral region in the complex plane and, within this region, organize themselves along a deformed rectangular lattice in the large degree limit. Our method of attack is a combination of the isomonodromic deformation method and a complex WKB approach to the biconfluent Heun equation.
This talk is based on two joint papers with Davide Masoero:
-Roots of generalized Hermite polynomials when both parameters are large, ArXiv, 2019;
-Poles of Painlev\'e IV Rationals and their Distribution, SIGMA, 2018.

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