A Monotonicity Result for the Range of a Perturbed Random Walk

We consider a discrete time simple symmetric random walk on $$\mathbb{Z }^d,\,d\ge 1,$$ where the path of the walk is perturbed by inserting deterministic jumps. We show that for any time $$n\in \mathbb{N }$$ and any deterministic jumps that we insert, the expected number of sites visited by the perturbed random walk up to time $$n$$ is always larger than or equal to that for the unperturbed walk. This intriguing problem arises from the study of a particle among a Poisson system of moving traps with sub-diffusive trap motion. In particular, our result implies a variant of the Pascal principle, which asserts that among all deterministic trajectories the particle can follow, the constant trajectory maximizes the particle’s survival probability up to any time $$t>0.$$