The continuum random tree. II: An overview. (English) Zbl 0791.60008

Aldous, David

Stochastic analysis, Proc. Symp., Durham/UK 1990, Lond. Math. Soc. Lect. Note Ser. 167, 23-70 (1991).

[For the entire collection see Zbl 0733.00017.]
[For part I see author, Ann. Probab. 19, No. 1, 1-28 (1991; Zbl 0722.60013).]
This paper is concerned with models of random trees on \(n\) vertices in which the average distance between pairs of vertices grows as order \(n^{1/2}\). The author presents an extensive review on old and recent results about random trees equivalent to Galton-Watson branching processes conditioned on the total population size equal to \(n\). Such models of trees, after appropriate rescaling the edges, converge in distribution as \(n \to \infty\) to the so-called compact continuum random tree which is closely connected with the Brownian excursion and the 3- dimensional Bessel process. Limit theorems of such type are discussed in detail. Finally, the author proposes models where the limit trees obtained are considered as variants of a superprocess. [For part III see below].

Reviewer: L.Mutafchiev (Sofia)

MSC:

60C05	Combinatorial probability
05C05	Trees
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
05C80	Random graphs (graph-theoretic aspects)

Keywords:

limit theorems; models of random trees; Galton-Watson branching processes; Brownian excursion; Bessel process