Critical branching random walks with small drift

We study critical branching random walks (BRWs) U(n)U(n) on Z+Z+ where the displacement of an offspring from its parent has drift 2β/n towards the origin and reflection at the origin. We prove that for any α>1α>1, conditional on survival to generation [nα][nα], the maximal displacement is ∼(α−1)/(4β)nlogn. We further show that for a sequence of critical BRWs with such displacement distributions, if the number of initial particles grows like ynαynα for some y>0y>0, α>1α>1, and the particles are concentrated in [0,O(n)], then the measure-valued processes associated with the BRWs converge to a measure-valued process, which, at any time t>0t>0, distributes its mass over R+R+ like an exponential distribution.