The Non-Orientable Map Asymptotics Constant p_g

2014/06/06 by Carrell, S. R.
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1406.1760

Abstract

Using the pfaffian structure of the generating series for locally orientable maps, we show that the generating series satsifies a nonlinear differential equation called the BKP equation. Using this we are able to derive a cubic differential equation which is satisfied by the generating series for locally orientable triangulations. As a result, we prove a conjecture of Garoufalidis and Mariño concerning the constant p_g which appears in asymptotic formulas for a variety of rooted maps on non-orientable surfaces. This allows one to determine the asymptotic expansion for p_g up to an unknown Stokes constant.

Related

• Universality and asymptotics of graph counting problems in nonorientable surfaces
• The KP hierarchy, branched covers, and triangulations
• Differential equations for the series of hypermaps with control on their full degree profile
• A direct method to find Stokes multipliers in closed form for P1 and more general integrable systems
• Full Asymptotic Expansion of Monodromy Data for the First Painlevé Transcendent: Applications to Connection Problems