2005/03/14 by Jean-Francois Le Gall, Gall, Jean-Francois Le
Mathematics · #60F17 #60J80 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F17 #msc:60J80
paper · pdf · doi:10.48550/arxiv.math/0503263
30 pages
arxiv created 2005/03/14 · arxiv updated 2009/12/01
We consider Galton-Watson trees associated with a critical offspring distribution and conditioned to have exactly n vertices. These trees are embedded in the real line by affecting spatial positions to the vertices, in such a way that the increments of the spatial positions along edges of the tree are independent variables distributed according to a symmetric probability distribution on the real line. We then condition on the event that all spatial positions are nonnegative. Under suitable assumptions on the offspring distribution and the spatial displacements, we prove that these conditioned spatial trees converge as n→∞, modulo an appropriate rescaling, towards the conditioned Brownian tree that was studied in previous work. Applications are given to asymptotics for random quadrangulations.