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Galton-Watson trees with vanishing martingale limit

Berestycki, N; Gantert, N; Morters, P; Sidorova, N; (2012) Galton-Watson trees with vanishing martingale limit.

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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.

Type:Article
Title:Galton-Watson trees with vanishing martingale limit
Additional information: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
UCL classification:UCL > School of BEAMS > Faculty of Maths and Physical Sciences > Mathematics

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