Universality and asymptotics of graph counting problems in nonorientable surfaces

Understand

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.