Discrete Painlevé Equations, Orthogonal Polynomials, and Counting Maps

Abstract

We analyze the recurrence coefficients for orthogonal polynomials on the real line. These orthogonal polynomials are defined by integration against the exponential of an even degree polynomial. With these weights of integration being exponentials of polynomials, the recurrence coefficients satisfy what is known as Freud's equation [Fre76]. Taking a dynamical systems perspective, we view Freud's equation as defining a non-autonomous discrete-time dynamics on the plane. In the case where the weight is the exponential of an even quartic, this dynamical system is an instance of a discrete Painlevé I equation [Mag99]. We approximate the orbits of this Painlevé equation using high precision arithmetic, and analyze the numerical sensitivity in plotting the orbits of this system. The orbit given by the orthogonal polynomial recurrence coefficients is shown to be tending towards a fixed point at infinity, along a center direction. Using an approximation of this center manifold, we provide estimates of the recurrence coefficients. Bauldry, Máté, and Nevai proved the existence of an asymptotic expansion for these recurrence coefficients in [BMN88]. We provide a recursion for computing the coefficients of this expansion. Using this recursion, together with several rescaling arguments, we relate this expansion to the asymptotic expansion proved by Ercolani, McLaughlin, and Pierce in [EMP08]. This connection enables us to find closed forms for the generating functions of 4-valent 2-legged labeled maps on a genus g surface, for any genus. We provide these generating functions for genera 0 through 3.