A pattern of asymptotic vertex valency distributions in planar maps

Let a vertex be selected at random in a set of n-edged rooted planar maps and p(k) denote the limit probability (as n --> infinity) of this vertex to be of valency k. 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-zero p(k) behave asymptotically in a uniform manner: p(k) similar to c (pi k)(-1/2) r(k) as k --> infinity With some constants I and c depending on the class. This distribution pattern can be reformulated in terms of the root vertex valency. By contrast, p(k) = 2(-k) for the class of arbitrary plane trees and p(k) = (k - 1)2(-k) for triangular dissections of convex polygons. (C) 1999 Academic Press.