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Title of Journal: J Stat Phys

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Abbravation: Journal of Statistical Physics

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Springer US

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DOI

10.1016/0360-1323(80)90035-9

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1572-9613

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Galton–Watson Trees with Vanishing Martingale Limi

Authors: Nathanaël Berestycki Nina Gantert Peter Mörters Nadia Sidorova
Publish Date: 2014/03/23
Volume: 155, Issue: 4, Pages: 737-762
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Abstract

We show that an infinite Galton–Watson tree conditioned on its martingale limit being smaller than varepsilon 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/varepsilon More precisely we show that if mu ge 2 then with high probability as varepsilon 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 Our proofs are based on recent results on the left tail behaviour of the martingale limit obtained by Fleischmann and Wachtel 11

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