Galton-Watson trees with vanishing martingale limit

Abstract

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 $\mu$-ary tree, where $\mu$ 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 $\mu\ge 2$ 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 $\mu$-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy.Comment: 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 analysi