Asymptotics and Dynamics of Map Enumeration Problems

Abstract

We solve certain three-term recurrence relations for generating functions of map enumeration problems. These are combinatorial maps, an embedding of a graph into a surface in a particular way. The generating functions enumerate the maps according to an appropriate notion of a distance or height in the map. These problems were studied and the recurrence relations derived in [BDFG03] and [BM06].By viewing the three-term recurrence as giving a two-dimensional discrete dynamical system, these combinatorial problems are set in the context of discrete dynamical systems and integrable systems theory. The integrable nature of the system was made apparent by numerical study, and is confirmed by recognition that the recurrences are autonomous discrete Painleve-I equations. The autonomous discrete Painleve equations are known to be instances of the QRT Mapping, named for Quispel, Roberts, and Thompson [QRT88, QRT89], an integrable structure with explicitly-given invariant. Level sets of such invariants are in general elliptic curves, and thus orbits in the dynamical systems can be parametrized through elliptic functions. The solution to a recurrence relation for combinatorial generating functions is rigorously derived from the general elliptic parametrization of the dynamical system, as the combinatorial initial condition indicates that the combinatorial orbit actually lies on a stable manifold of a hyperbolic fixed point of the system. This special orbit thus lies on a separatrix of the system, which is given by a degeneration in the elliptic nature of the level sets of the invariant function. These solutions have a particularly nice algebraic form, which is seen to be a consequence of the degeneration of the elliptic parametrization. The framework and method are general, applicable to any combinatorial enumeration problem that arises with a similar QRT-type structure.