Subdiffusivity of a random walk among a Poisson system of moving traps on ${\mathbb Z}$

TL;DR

This paper demonstrates that a random walk conditioned on survival among moving traps on the integer lattice exhibits subdiffusive behavior, providing new bounds on thin points and holes in the walk's range.

Contribution

It establishes the subdiffusivity of the walk conditioned on survival and derives bounds on thin points and holes, advancing understanding of random walks in dynamic random environments.

Findings

Random walk conditioned on survival is subdiffusive.

Provides an upper bound on the number of thin points.

Bounds the total volume of holes in the walk's range.

Abstract

We consider a random walk among a Poisson system of moving traps on ${\mathbb Z}$. In earlier work [DGRS12], the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time $t$ in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk's range.