Critical Branching Random Walks with Small Drift

We study critical branching random walks (BRWs) $U^{(n)}$ on~$\mathbb{Z}_{+}$ where for each $n$, the displacement of an offspring from its parent has drift~$2β/\sqrt{n}$ 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 asymptotically equivalent to $(α-1)/(4β)\sqrt{n}\log n$. We further show that for a sequence of critical BRWs with such displacement distributions, if the number of initial particles grows like~$yn^α$ for some $y>0$ and $α>1$, and the particles are concentrated in~$[0,O(\sqrt{n})],$ then the measure-valued processes associated with the BRWs, under suitable scaling converge to a measure-valued process, which, at any time~$t>0,$ distributes its mass over~$\mathbb{R}_+$ like an exponential distribution.

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