The Asymptotic Behavior of a Random Walk Among a Field of Traps

Abstract

We consider the problem of random walks
moving around on a lattice Zd with an initial Poisson distribution of traps. We
consider both static and moving traps. In the static case, we prove that the
survival time has a decay of e−c t d /d +2 based on a heuristic argument. In
the moving case we aim to prove the sub-exponential decay of the survival time
in dimensions 1 and 2, as well as the exponential de- cay of survival in
dimensions 3 and higher. We achieve the former by first expressing the survival
probability in the range of a random walk and by showing that the asymptotic
behavior of said range behaves in a sub-exponential and exponential way for
dimensions 1/2 and ≥ 3 respectively. Further- more, we also show an upper bound
for the survival time of the form lim supt →∞ 1 t log P(T ≥ t ) < 0. Following
this we look at the situation where traps decay as ||x|| → ∞. Meaning less
traps will be distributed further away from the origin. We show that in the
static case, if the decay rate satisfies the condition px ≤ p(||x||) where p(r
) is non-increasing and r p(r ) is integrable and convergent, that the random
walk will be transient, meaning that there will be a strictly positive chance
of survival. Lastly, we then show that for the dynamically moving traps case,
if the decay rate is "fast enough", meaning that if the Poisson
parameter of the distribution of the traps ρ(x) is of the form 1/||x||2+α where
α > d − 2, that there will also be a strictly positive probability of
survival.