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Other version: http://linkinghub.elsevier.com/retrieve/pii/S009731650900171X
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Universality and asymptotics of graph counting problems in nonorientable surfaces☆ 

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Published in  Journal of Combinatorial Theory  Series A. 2010, vol. 117, no. 6, p. 715  740  
Abstract  Bender–Canfield showed that a plethora of graph counting problems in orientable/nonorientable surfaces involve two constants tg and pg for the orientable and the nonorientable case, respectively. T.T.Q. Le and the authors recently discovered a hidden relation between the sequence tg and a formal power series solution u(z) of the Painlevé I equation which, among other things, allows to give exact asymptotic expansion of tg 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 pg 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 3manifold 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.  
Stable URL  http://archiveouverte.unige.ch/unige:11879  
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