Height and diameter of Brownian tree. (English) Zbl 1335.60162

Wang, Minmin

Electron. Commun. Probab. 20, Paper No. 88, 15 p. (2015).

Summary: By computations on generating functions, G. Szekeres [Lect. Notes Math. 1036, 392–397 (1983; Zbl 0524.05026)] proved in 1983 that the law of the diameter of a uniformly distributed rooted labelled tree with \(n\) vertices, rescaled by a factor \(n^{-1/2}\), converges to a distribution whose density is explicit. D. Aldous [Ann. Probab. 19, No. 1, 1–28 (1991; Zbl 0722.60013)] observed in 1991 that this limiting distribution is the law of the diameter of the Brownian tree. In our article, we provide a computation of this law which is directly based on the normalized Brownian excursion. Moreover, we provide an explicit formula for the joint law of the height and diameter of the Brownian tree, which is a new result.

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J65	Brownian motion

Keywords:

Brownian tree; Brownian excursion; continuum random tree; Jacobi theta function; Williams decomposition

Citations:

Zbl 0524.05026; Zbl 0722.60013