Random walk through a fertile site

We study the dynamics of random walks hopping on homogeneous hypercubic lattices and multiplying at a fertile site. In one and two dimensions, the total number N(t) of walkers grows exponentially at a Malthusian rate depending on the dimensionality and the multiplication rate mu at the fertile site. When d > d(c) = 2, the number of walkers may remain finite forever for any mu; it surely remains finite when mu <= mu(d). We determine mu(d) and show that < N(t)> grows exponentially if mu > mu(d). The distribution of the total number of walkers remains broad when d <= 2, and also when d > 2 and mu > mu(d). We compute < N-m > explicitly for small m, and show how to determine higher moments. In the critical regime, < N > grows as root t for d = 3, t/ In t for d = 4, and t for d > 4. Higher moments grow anomalously, < N-m > similar to < N >(2m-1), in the critical regime; the growth is normal, < N-m > similar to < N >(m), in the exponential phase. The distribution of the number of walkers in the critical regime is asymptotically stationary and universal, viz., it is independent of the spatial dimension. Interactions between walkers may drastically change the behavior. For random walks with exclusion, if d > 2, there is again a critical multiplication rate, above which < N(t)> grows linearly (not exponentially) in time; when d <= d(c) = 2, the leading behavior is independent on mu and < N(t)> exhibits a sublinear growth.

Automated summary

The study investigates the dynamics of random walks on homogeneous hypercubic lattices with multiplication at a fertile site, finding that the total number of walkers grows exponentially at a Malthusian rate in one and two dimensions. The behavior of walker numbers is determined by dimensionality and multiplication rate, with a critical regime showing anomalous growth. Interactions between walkers can significantly alter their behavior, with exclusion random walks exhibiting linear growth above a critical multiplication rate.