A conditioning principle for Galton-Watson trees

Abstract

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than \eps, converges as \eps\downarrow 0 in law to the regular $\mu$-ary tree, where $\mu$ is the essential minimum of the offspring distribution. This gives an example of entropic repulsion where the limit has no entropy.Comment: This is now superseded by a new paper, arXiv:1204.3080, written jointly with Nina Gantert. The new paper contains much stronger results (e.g. the two-point concentration of the level at which the Galton-Watson tree ceases to be minimal) based on a significantly more delicate analysis, making the present paper redundan