Galton-Watson trees with vanishing martingale limit

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

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