Random walk in a dynamic environment of traps

Abstract

Various problems related to Random Walk in Random Environments have been researched intensively over the past 40 years by both the mathematics and physics communities. The setup of these problems is quite simple: one defines a Random Walk of choice and allows it to traverse a Random Environment that influences the Random Walk in a predetermined way.

In this thesis, we study the survival probability of a Random Walk in various random environments of traps. These Random Environments of traps are divided into two distinct classes: static and dynamic. In the static environment, the trap configuration is randomly chosen at time zero and remains fixed thereafter. In the dynamic case, the trap configuration evolves over time according to some predefined dynamics.

The problem in the static setting has a long history, and the situation is fairly well understood. For the dynamic environments, however, the state of the art is less developed, as there is no unified approach to deriving survival probabilities for different dynamic environments. Additionally, almost nothing is known about the interpolation between the models of random walk in static and dynamic environments. The interpolation model should exhibit exponential decay of survival probability in a fast environment, corresponding to the dynamic environment, and show subexponential decay of survival probability when the environment is slow, corresponding to the static environment.

One of the goals of this thesis is to compile various results of survival probabilities in different dynamic environments and to create a model that allows interpolation between exponential and subexponential decay of survival probability.