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Báo cáo toán học: "he map asymptotics constant tg"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The map asymptotics constant tg. | The map asymptotics constant tg Edward A. Bender Department of Mathematics University of California San Diego La Jolla CA 92093-0112 ebender@ucsd.edu Zhicheng Gao School of Mathematics and Statistics Carleton University Ottawa Ontario K1S5B6 Canada zgao@math.carleton.ca L. Bruce Richmond Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario N2L 3G1 Canada Submitted Jan 28 2008 Accepted Mar 22 2008 Published Mar 27 2008 Mathematics Subject Classification 05C30 Abstract The constant tg appears in the asymptotic formulas for a variety of rooted maps on the orientable surface of genus g. Heretofore studying this constant has been difficult. A new recursion derived by Goulden and Jackson for rooted cubic maps provides a much simpler recursion for tg that leads to estimates for its asymptotics. 1 Introduction Let Sg be the orientable surface of genus g. A map on Sg is a graph G embedded on Sg such that all components of Sg G are simply connected regions. These components are called faces of the map. A map is rooted by distinguishing an edge an end vertex of the edge and a side of the edge. With Mn g the number of rooted maps on Sg with n edges Bender and Canfield 1 showed that Mn g tgn5 g 1 212n as n 1 1 Research supported by NSERC y Research supported by NSERC THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R51 1 where the tg are positive constants which can be calculated recursively using a complicated recursion involving in addition to g many other parameters. The first three values are 2 1 7 0 1 24 and 2 43207F Gao 3 showed that many other interesting families of maps also satisfy asymptotic formulas of the form atg pn 5 g 1 27n 2 and presented a table of n p and 7 for eleven families. Richmond and Wormald 5 showed that many families of unrooted maps have asymptotics that differ from the rooted asymptotics by a factor of four times the number of edges. See Goulden and Jackson 4 for a discussion of connections with mathematical physics.