Probabilistic aspects of infinite trees and surfaces

We present a simple construction of a probability measure on rooted infinite planar trees as a limit of a sequence of uniform measures on finite trees. We compute the conditional probability measure on the set of trees containing a given finite tree and use this to determine the distribution of the number of vertices at a given distance from the root, and thereby the Hausdorff dimension associated with this measure. The construction can be generalised to other ensembles of infinite discrete structures. We indicate, in particular, how it can be adapted in a straight forward manner to obtain a probability measure on infinite planar surfaces by using a certain correspondence between quadrangulated surfaces and so-called well labelled trees. The Hausdorff dimension of this measure turns out to be 4. Details of these latter results will appear elsewhere.