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

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

Benjert, Zylan (TU Delft Electrical Engineering, Mathematics and Computer Science)

Redig, F.H.J. (mentor)
Janssens, B. (graduation committee)

Delft University of Technology

Applied Mathematics

2023-07-20

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.

Random Walk
Poisson
Traps
Survival Probability
Decay

http://resolver.tudelft.nl/uuid:9b256cc1-0267-43f3-bfea-16de4e6d9f00

Student theses

bachelor thesis

© 2023 Zylan Benjert