Critical branching random walks with small drift

We study critical branching random walks (BRWs) U ( n ) on  Z + where the displacement of an offspring from its parent has drift  2 β / n towards the origin and reflection at the origin. We prove that for any  α > 1 , conditional on survival to generation  [ n α ] , the maximal displacement is ∼ ( α − 1 ) / ( 4 β ) n log n . We further show that for a sequence of critical BRWs with such displacement distributions, if the number of initial particles grows like  y n α for some y > 0 ,  α > 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 > 0 , distributes its mass over  R + like an exponential distribution.