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(πk)−1/2 rkask→∞ 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.