# Large-degree asymptotics of rational Painleve-IV functions associated to generalized Hermite polynomials

## TL;DR

This paper analyzes the asymptotic behavior of rational solutions to the Painleve-IV equation generated by generalized Hermite polynomials, revealing their zero and pole distributions in the limit of large indices.

## Contribution

It extends the asymptotic analysis of Painleve-IV functions by expressing them via orthogonal polynomials and applying the Deift-Zhou steepest descent method for large parameter limits.

## Key findings

- Zeros and poles densely fill curvilinear rectangles asymptotically.
- Explicit boundary curve for the zero-pole distribution is derived.
- Leading-order asymptotics of the functions in pole-free regions are obtained.

## Abstract

The Painleve-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers m and n. These functions have applications to nonlinear wave equations, random matrices, fluid dynamics, and quantum mechanics. Numerical studies suggest the zeros and poles form a deformed n by m rectangular grid. Properly scaled, the zeros and poles appear to densely fill certain curvilinear rectangles as m and n tend to infinity with r=m/n fixed. Generalizing a method of Bertola and Bothner used to study rational Painleve-II functions, we express the generalized Hermite rational Painleve-IV functions in terms of certain orthogonal polynomials on the unit circle. Using the Deift-Zhou nonlinear steepest-descent method, we asymptotically analyze the associated Riemann-Hilbert problem in the limit as n tends to infinity with m=r*n for r fixed. We obtain an explicit characterization of the boundary curve and determine the leading-order asymptotic expansion of the functions in the pole-free region.