Universality and asymptotics of graph counting problems in nonorientable surfaces

Bender-Canfield showed that a plethora of graph counting problems in oriented/unoriented surfaces involve two constants $t_g$ and $p_g$ for the oriented and the unoriented 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 Painlev\'e 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 nonlinear 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 $\mathrm{O}(N)$ and $\mathrm{Sp}(N)$-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.