Universality and asymptotics of graph counting problems in nonorientable surfaces

Details Universality and asymptotics of graph counting problems in nonorientable surfaces Universality and asymptotics of graph counting problems in nonorientable surfaces

2008-12-05

arxiv.org/abs/0812.1195v4

Mathematics

Combinatorics

24 pages and 5 figures

Scientific paper

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.

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