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

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• 作者：Stavros Garoufalidis ; Marcos Mariñ ; o
• 关键词：Rooted maps ; Non-orientable surfaces ; Cubic ribbon graphs ; Quadrangulations ; Stokes constants ; Painlevé ; I asymptotics ; Riemann– ; Hilbert method ; Borel transform ; Trans-series ; Matrix models ; Double-scaling limit ; Instantons
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2010
• 出版时间：August 2010
• 年：2010
• 卷：117
• 期：6
• 页码：715-740
• 全文大小：498 K

文摘

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.