A Monotonicity Result for the Range of a Perturbed Random Walk

TL;DR

This paper proves that inserting deterministic jumps into a symmetric random walk increases the expected number of visited sites, and it establishes a variant of the Pascal principle related to particle survival among moving traps.

Contribution

It demonstrates a monotonicity property of the range of a perturbed random walk and links this to a survival probability maximization principle.

Findings

Expected visited sites increase with deterministic jumps.

Constant trajectory maximizes particle survival probability.

Monotonicity holds for any dimension d>=1.

Abstract

We consider a discrete time simple symmetric random walk on Z^d, d>=1, where the path of the walk is perturbed by inserting deterministic jumps. We show that for any time 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.