A Pattern of Asymptotic Vertex Valency Distributions in Planar Maps

摘要: Let a vertex be selected at random in set ofn-edged rooted planar maps andpkdenote the limit probability (asn?∞) of this to valencyk. For diverse classes including Eulerian, arbitrary, polyhedral, and loopless as well 2- 3-connected triangulations, it is shown that non-zeropkbehave asymptotically auniformmanner:pk~c(?k)?1/2rkask?∞ with some constantsrandcdepending on class. This distribution pattern can reformulated terms root valency. By contrast,pk=2?kfor class arbitrary plane trees andpk=(k?1)2?kfor triangular dissections convex polygons.