Random Walking: Ants to Electrons

Consider an ant who starts crawling, one step at a time, choosing a direction at random for each step. More specifically, we would consider that the ant has a certain chance to step in finite number of directions. This is an example of a random walk on a graph. One can ask questions such as: If there is both a food source and a trap, what is the probability that the ant will reach the food source before the trap? How many times would the ant be expected to retrace its steps? How many steps could the ant expect to take to reach the food source? Could the random path of a hungry ant somehow be related to that of an electron in an electric circuit?

Random walks are a type of Markov chain, which is any random process in which each step only depends on the last step in the sequence. In this project, students will explore the answer to some of the questions posed above and learn about some of properties of random walks and Markov chains in general. Then the students will use random walks to model real world phenomenon including elementary electric networks. Students will also use Matlab to run simulations of random walks.

For this project students should be familiar with mathematical concepts from Math 31. A familiarity with the basic rules of probability and the basics of a programming language such as Matlab or C++ would be helpful but not required.