Slumpvandringar i slumpmässiga miljöer - En simuleringsbaserad undersökning av rekurrens och transiens

Abstract

The term “random walk in random environment” can have several different meanings. In this study, random walks in three different types of environments will be examined: a non-random environment, used as a reference point, and two variations of random environments referred to as a static random environment and a dynamic random environment.

The study investigates random walks in one- and two dimensions. Random walks in d=1 can described as occurring on an integer line, while in d=2 on an integer grid.

Random walks can be described by their long-term behavior where recurrence refers to how the random walker tends to return to the origin while transience refers to how the random walker moves away from the origin without returning.

Theoretical results for recurrence and transience have been derived in all dimensions for the non-random environment, as well for the static random environment in d=1. However, for random walks in random environments in d=2 general theoretical results for these behaviors have not yet been derived.

This study uses a simulationbased method to investigate how bias affects recurrence and transience in dimensions d=1 and d=2 across different environment types. The results showed that dimension has a great influence on tendencies for these behaviors. Under symmetry, i.e. at bias level 0, random walks in the non-random and dynamic random environment held similar results. In the static random environment certain areas affected how the random walker was able to move. The results also showed that as the level of bias increased, the movement became more directed, and the effect from the bias took over.