The depth first processes of Galton--Watson trees converge to the same Brownian excursion

Abstract

In this paper, we show a strong relation between the depth first processes associated to Galton-Watson trees with finite variance, conditioned by the total progeny: the depth first walk, the depth first queue process, the height process; a consequence is that these processes (suitably normalized) converge to the same Brownian excursion. This provides an alternative proof of Aldous' one of the convergence of the depth first walk to the Brownian excursion which does not use the existence of a limit tree. The methods that we introduce allow one to compute some functionals of trees or discrete excursions; for example, we compute the limit law of the process of the height of nodes with a given out-degree, and the process of the height of nodes, root of a given subtree.

Recent progress in coalescent theory

“…The result now follows. Further details can be found in Aldous [3], but see also Marckert and Mokkadem [117] and Le Gall and Le Jan [114].…”

Vertices with fixed outdegrees in large Galton-Watson trees

“…This result is due to Marckert and Mokkadem [28] under the assumption that µ has a finite exponential moment and to Duquesne [10] for the general case.…”

“…The first one is a joint scaled convergence in distribution of the contour process (which was defined in the Introduction) and the Lukasiewicz path of a critical GW tree with finite variance, conditioned to have n vertices, to the same Brownian excursion. Theorem 2.2 (Marckert and Mokkadem [28], Duquesne [10]). Let µ be a critical offspring distribution with finite positive variance σ 2 .…”

Epidemics on critical random graphs with heavy-tailed degree distribution

“…Using the results in the literature on Galton-Watson trees conditioned on having a fixed size [50,36,54,7] one can recover the Jeulin identity [45] and its α-stable extension due to Miermont [56] from our Theorem 4 as well. The proofs in [56,45] do not rely on weak convergence arguments.…”