Universality and asymptotics of graph counting problems in non-orientable surfaces

Bender–Canfield showed that a plethora of graph counting problems in orientable/non-orientable surfaces involve two constants t g and p g for the orientable and the non-orientable case, respectively. T.T.Q. Le and the authors recently discovered a hidden relation between the sequence t g and a formal power series solution u ( z ) of the Painlevé I equation which, among other things, allows to give exact asymptotic expansion of t g to all orders in 1 / g for large g. The paper introduces a formal power series solution v ( z ) of a Riccati equation, gives a non-linear recursion for its coefficients and an exact asymptotic expansion to all orders in g for large g, using the theory of Borel transforms. In addition, we conjecture a precise relation between the sequence p g and v ( z ) . Our conjecture is motivated by the enumerative aspects of a quartic matrix model for real symmetric matrices, and the analytic properties of its double scaling limit. In particular, the matrix model provides a computation of the number of rooted quadrangulations in the 2-dimensional projective plane. Our conjecture implies analyticity of the O ( N ) - and Sp ( N ) -types of free energy of an arbitrary closed 3-manifold in a neighborhood of zero. Finally, we give a matrix model calculation of the Stokes constants, pose several problems that can be answered by the Riemann–Hilbert approach, and provide ample numerical evidence for our results.