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 \(t>0.\)