Galton-watson Trees with Vanishing Martingale Limit

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than ε, 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/ε). More precisely, we show that if μ ≥ 2 then with high probability as ε ↓ 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. Mathematics Subject Classification (2010): 60J80 (Primary) 60F10, 60K37.