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Maps are polygonal cellular networks on Riemann surfaces. This paper analyzesthe construction of closed form general representations for the enumerativegenerating functions associated to maps of fixed but arbitrary genus. Themethod of construction developed here involves a novel asymptotic symbolcalculus for difference operators based on the relation between spectralasymptotics for Hermitian random matrices and asymptotics of orthogonalpolynomials with exponential weights. These closed form expressions have auniversal character in the sense that they are independent of the explicitvalence distribution of the cellular networks within a broad class.Nevertheless the valence distributions may be recovered from the closed formgenerating functions by a remarkable unwinding identity in terms of Appellpolynomials generated by Bessel functions. Our treatment reveals the generatingfunctions to be solutions of nonlinear conservation laws and theirprolongations. This characterization enables one to gain insights that gobeyond more traditional methods that are purely combinatorial. Universalityresults are connected to stability results for characteristic singularities ofconservation laws that were studied by Caflisch, Ercolani, Hou and Landis aswell as directly related to universality results for random matrix spectra.

Relating random matrix map enumeration to a universal symbol calculus for recurrence operators in terms of Bessel-Appell polynomials