Random walk through a fertile site

Michel Bauer, P. L. Krapivsky, Kirone Mallick

Published 2019-07-30Version 1

We study non-interacting random walkers (RWs) on homogeneous hyper-cubic lattices with one special fertile site where RWs can reproduce at rate $\mu$. We show that the total number $\mathcal{N}(t)$ and the density of RWs at any site grow exponentially with time in low dimensions, $d=1$ and $d=2$; above the lower critical dimension, $d>d_c=2$, the number of RWs may remain finite forever for any $\mu$, and surely remains finite when $\mu\leq \mu_d$. We determine the critical multiplication rate $\mu_d$ and show that the average number of RWs grows exponentially if $\mu>\mu_d$. The distribution $P_N(t)$ of the total number of RWs remains broad when $d\leq 2$, and also when $d>2$ and $\mu>\mu_d$. We derive explicit expressions for the first moments of $\mathcal{N}(t)$ and establish a recurrence that allows, in principle, to compute an arbitrary moment. In the critical regime, $\langle \mathcal{N}\rangle$ grows as $\sqrt{t}$ for $d=3$, $t/\ln t$ for $d=4$ and $t$ (for $d>4$). Higher moments grow anomalously, $\langle \mathcal{N}^m\rangle\sim \langle \mathcal{N}\rangle^{2m-1}$, instead of the normal growth, $\langle \mathcal{N}^m\rangle\sim \langle \mathcal{N}\rangle^{m}$, valid in the exponential phase. The distribution of the number of RWs in the critical regime is asymptotically stationary and universal, viz. it is independent of the spatial dimension.