Critical Branching Random Walks with Small Drift

Abstract

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~$2\beta /\sqrt{n}$ towards the origin and reflection at the origin. We prove that for any~$\alpha >1$, conditional on survival to generation~$[n^{\alpha }]$, the maximal displacement is asymptotically equivalent to $(\alpha -1)/(4\beta )\sqrt{n}\log n$. We further show that for a sequence of critical BRWs with such displacement distributions, if the number of initial particles grows like~$y{n}^{\alpha }$ for some $y>0$ and $\alpha >1$, 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.

References in corpus (1)

• Occupation Statistics of Critical Branching Random Walks in Two or Higher Dimensions