Range-controlled random walks

We introduce range-controlled random walks with hopping rates depending on the range $\mathcal{N}$, that is, the total number of previously distinct visited sites. We analyze a one-parameter class of models with a hopping rate $\mathcal{N}^a$ and determine the large time behavior of the average range, as well as its complete distribution in two limit cases. We find that the behavior drastically changes depending on whether the exponent $a$ is smaller, equal, or larger than the critical value, $a_d$, depending only on the spatial dimension $d$. When $a>a_d$, the forager covers the infinite lattice in a finite time. The critical exponent is $a_1=2$ and $a_d=1$ when $d\geq 2$. We also consider the case of two foragers who compete for food, with hopping rates depending on the number of sites each visited before the other. Surprising behaviors occur in 1d where a single walker dominates and finds most of the sites when $a>1$, while for $a<1$, the walkers evenly explore the line. We compute the gain of efficiency in visiting sites by adding one walker.