Title:Rational solutions of Painlevé V from Hankel determinants and the asymptotics of their pole locations

Abstract:In this paper we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlevé equation. These solutions are constructed by relating the corresponding tau function with a Hankel determinant of a certain sequence of moments. This approach was also used by one of the authors and collaborators in the study of the rational solutions of the second Painlevé equation. More specifically we study the roots of the corresponding polynomial tau function, whose location corresponds to the poles of the associated rational solution. We show that, upon suitable rescaling, the roots fill a well-defined region bounded by analytic arcs when the degree of the polynomial tau function tends to infinity. Moreover we provide an approximate location of these roots within the region in terms of suitable quantization conditions.