TAILIEUCHUNG - Báo cáo toán học: "A formula for the bivariate map asymptotics constants in terms of the univariate map asymptotics constants"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A formula for the bivariate map asymptotics constants in terms of the univariate map asymptotics constants. | A formula for the bivariate map asymptotics constants in terms of the univariate map asymptotics constants Zhicheng Gao School of Mathematics and Statistics Carleton University Ottawa Canada K1S 5B6 Submitted Oct 18 2010 Accepted Nov 9 2010 Published Nov 19 2010 Mathematics Subject Classification 05C10 05C30 Abstract The parameters tg pg tg r and pg r appear in the asymptotics for a variety of maps on surfaces and embeddable graphs. In this paper we express tg r in terms of tg and pg r in terms of pg. 1 Introduction The concepts in this paragraph will be made precise in the following paragraphs. The parameters tg and pg arise in the univariate asymptotic enumeration of a variety of maps on surfaces and the parameters tg r and pg r arise in the corresponding bivariate asymptotics for maps as well as embeddable graphs. The original recursions for these parameters make it extremely difficult to compute them for higher genus surfaces. In contrast the other parameters in the asymptotics are usually easily determined. Recently a simple recursion has been obtained for tg and another conjectured for pg. In this paper we obtain simple expressions for the bivariate parameters tg r and pg r in terms of the corresponding univariate parameters. A map is a connected graph G embedded in a surface S a closed 2-manifold such that all components of S G are simply connected regions which are called faces. Loops and multiple edges are allowed in G. A map is rooted if an edge is distinguished together with a direction on the edge and a side of the edge. The exact enumeration of various types of maps on the sphere or equivalently the plane was carried out by Tutte and his students see 28 for a survey in the 1960s via his device of rooting. Beginning in the 1980s Tutte s approach was used for the asymptotic enumeration of maps on general surfaces 3 4 9 11 16 17 18 19 . A matrix integral approach was initiated by t Research supported by NSERC THE ELECTRONIC JOURNAL OF COMBINATORICS 17 .

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