2012/04/13 by Nathanael Berestycki, Nina Gantert, Berestycki, Nathanael +5 · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1204.3080
This supersedes an earlier paper, arXiv:1006.2315, written by a subset of the authors. Compared with the earlier version, the main result (the two-point concentration of the level at which the Galton-Watson tree ceases to be minimal) is much stronger and requires significantly more delicate analysis
arxiv created 2012/04/13 · arxiv updated 2012/04/16
We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than \eps, agrees up to generation K with a regular μ-ary tree, where μ is the essential minimum of the offspring distribution and the random variable K is strongly concentrated near an explicit deterministic function growing like a multiple of log(1/\eps). More precisely, we show that if μ≥ 2 then with high probability as \eps \downarrow 0, K takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular μ-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy.