Galton–Watson Trees with Vanishing Martingale Limit

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].