Geodesic
The asymptotic number of maps on compact orientable surfaces
Diskretnaya Matematika, Tome 13 (2001) no. 2, pp. 89-98

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We get an asymptotic formula for the sum $$ Z_{N}=\sum_{b+p=N}F_{b,p}y^p, $$ where $$ F_{b,p}=\sum_{\rho=0}^\infty F_{b,p}(\rho), $$ and $F_{b,p}(\rho)$ is the number of maps of genus $\rho$ with $p+1$ vertices and $p+b$ edges. 
@article{DM_2001_13_2_a3,
     author = {M. A. Krikun and V. A. Malyshev},
     title = {The asymptotic number of maps on compact orientable surfaces},
     journal = {Diskretnaya Matematika},
     pages = {89--98},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2001_13_2_a3/}
}
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M. A. Krikun; V. A. Malyshev. The asymptotic number of maps on compact orientable surfaces. Diskretnaya Matematika, Tome 13 (2001) no. 2, pp. 89-98. http://geodesic.mathdoc.fr/item/DM_2001_13_2_a3/