Title: Large-degree asymptotics of rational Painleve-IV functions associated to generalized Hermite polynomials,
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.