不可定向曲面上图计数问题的普适性和渐近性
Universality and asymptotics of graph counting problems in nonorientable surfaces
摘要：
Bender-Canfield证明了有向/无向曲面中的大量图计数问题分别涉及有向和无向情况下的两个常数$t_g$和$p_g$。总智商Le和作者最近发现了序列$t_g$和Painlevé I方程的正式幂级数解$u(z)$之间的隐藏关系，其中，允许对大$g$的$1/g$中的所有阶进行$t_g$的精确渐近展开。利用Borel变换理论，给出了Riccati方程的形式幂级数解$v(z)$，给出了其系数的非线性递推式，并对大的$g$给出了$g$的精确渐近展开式.此外，我们还推测了序列$p_g$和$v(z)$之间的精确关系.我们的猜想是由实对称矩阵的四次矩阵模型的计数方面，其双重标度极限的分析性质的动机。特别地，矩阵模型提供了二维投影平面中的有根四边形的数量的计算。我们的猜想暗示了任意闭三维流形在零邻域内自由能的$\mathrm{O}(N)$和$\mathrm{Sp}(N)$型的解析性。最后，我们给出了一个矩阵模型计算的斯托克斯常数，提出了几个问题，可以回答的Riemann-Hilbert方法，并提供了充分的数值证据，我们的结果。
英文摘要：
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 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 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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