Galton-Watson trees with vanishing martingale limit

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than (Formula presented.), agrees up to generation (Formula presented.) with a regular (Formula presented.)-ary tree, where (Formula presented.) is the essential minimum of the offspring distribution and the random variable (Formula presented.) is strongly concentrated near an explicit deterministic function growing like a multiple of (Formula presented.). More precisely, we show that if (Formula presented.) then with high probability, as (Formula presented.), (Formula presented.) takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular (Formula presented.)-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 ].