A Pattern of Asymptotic Vertex Valency Distributions in Planar Maps

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

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