Stochastic Analysis: The Continuum random tree II: an overview

1 INTRODUCTION Many different models of random trees have arisen in a variety of applied setting, and there is a large but scattered literature on exact and asymptotic results for particular models. For several years I have been interested in what kinds of "general theory" (as opposed to ad hoc analysis of particular models) might be useful in studying asymptotics of random trees. In this paper, aimed at theoretical probabilists, I discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes. No prior knowledge of this subject is assumed: the paper is intended as an introduction and survey. To give the really big picture in a paragraph, consider a tree on n vertices. View the vertices as points in abstract (rather than d-dimensional) space, but let the edges have length (= 1, as a default) so that there is metric structure: the distance between two vertices is the length of the path between them. Consider the average distance between pairs of vertices. As n-> oo this average distance could stay bounded or could grow as order n, but almost all natural random trees fall into one of two categories. In the first (and larger) category, the average distance grows as order log n. This category includes supercritical branching processes, and most "Markovian growth" models such as those occurring in the analysis of algorithms. This paper is concerned with the second category, in which the average distance grows as order n1/2. This occurs with Galton-Watson branching processes conditioned on total population size = n (in brief, CBP(n)). At first sight that seems an unnatural model, but it turns out to coincide (see section 2.1) with various combina-torial models, and is similar to more general models of critical branching processes conditioned to be large (in any reasonable way). The fundamental Aldous: The continuum random tree II fact is that, by scaling edges to have length n-1/2, these random trees converge in distribution as n-+ oo to a limit we call the CCRT (for compact continuum random tree). This was treated explicitly in Aldous [2] in a special case and in Aldous [3] in the natural general case, though (as we shall see) many related results are implicit in recent literature. Thus asymptotic distributions for these models of discrete random trees can be obtained immediately from distributions associated with the limit tree. …