Random Walks : The Properties, Applications and Methods of Analysis

Random walks come in an array of interesting classes, each with unique properties, applications and methods of analysis. This paper will provide an analytical and numerical analysis of the different classes of random walks, and study the relationships that connect them. This paper will introduce and discuss the key concepts of simple random walks, Levy flights, reinforced random walks, self-avoiding walks, and Brownian motion. Following this, new research results will be presented. First, an array of numerical evidence will be introduced to support the Levy Flight Foraging Hypothesis. Further to this, upper and lower bounds for the connective constant of the Union Jack lattice will be implemented numerically. Additionally, a new method of analysis will be developed to study the Narrow Escape Problem. Lastly, an extension to the reinforced random walks will be constructed to link reinforced random walks to self-avoiding walks.