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

Published inJournal of combinatorial theory. Series A, vol. 117, no. 6, p. 715-740
Publication date2010
Abstract

Bender–Canfield showed that a plethora of graph counting problems in orientable/non-orientable surfaces involve two constants tg and pg for the orientable and the non-orientable 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 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 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 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.

Citation (ISO format)
GAROUFALIDIS, Stavros, MARINO BEIRAS, Marcos. Universality and asymptotics of graph counting problems in non-orientable surfaces☆. In: Journal of combinatorial theory. Series A, 2010, vol. 117, n° 6, p. 715–740. doi: 10.1016/j.jcta.2009.10.013
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ISSN of the journal0097-3165
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