Undergrad research project - spring 2019

Meeting time and place

The group will meet Fridays at 3:30 in ENR2 S375. The first meeting is Jan 11.

Topic

The ordinary random walk is defined as follows. Look at the square lattice Z^2 in two dimensions. We start at (0,0). At each time step we randomly pick one of the four directions (left, right, up, down) and take a step in that direction. There are many rigorous results for this walk. One example - the average distance the walk travels is proportional to sqrt(N) where N is the number of steps. If we call the step length delta and let delta go to zero, we get a random process called the scaling limit. For the ordinary random walk the scaling limit is Brownian motion. There is an amazing theorem that says this result is universal . For example, we could generalize the walk. At each time step we randomly choose a direction and we randomly choose the number of steps we will take in that direction (instead of just one). Under some mild assumptions, the scaling limit of all these random walk models will still be Brownian motion!

Now suppose we want to change the model so that it does not visit any site more than once. There are many way to do this, and the possible models have a fascinating varitey of behaviors. One simple model is to use the "Manhattan lattice". Each horizontal and vertical line gets a direction. The directions assigned to theses lines (the streets) alternate (like the streets in Manhattan). We now impose two constraints on the random walk: it must follow the directions of the "streets" and it is not allowed to traverse any section of street it has already traversed. This model has very different behavior. For example, the distance the walk travels is conjectured to be proportional to N^p where the power p is 4/7 instead of 1/2. The scaling limit of this model is believed to be the Schramm-Loewner evolution - a new stochastic process that was discovered around 2000. It appears there is a new form of universality at work here. Many different random walks that are not allowed to visit a site more than once seem to have the same scaling limit. I say "appears" and "seems" because this is a very active area of research and there are many unresolved conjectures.

Prerequisites and commitment

A very strong academic record in math courses. This project is intended for students who are doing very well in their math courses and are looking for a different kind of challenge. Students should know probability at the level of 464. We will do simulations, so some programming experience is a plus.
Students who do this for academic credit should expect to do at least as much work as they would for a 400 level math course.

Format

This will be a group project. We will work together to learn the necessary background and then work together on research projects. Here is a very tentative plan:

1. Warm-up: the ordinary random walk

The goal here is learn some background about the ordinary random walk. This will be relatively short.

2. Background - a non-intersecting random walk

The goal here is to work through this paper.

3. Research:

The goal here is to pose and try to answer new questions about the random walk in 2. Some possible directions:

Joining the project

If you are interested, send me an email (tgk@math.arizona.edu) ASAP. You should include a copy of your transcript and any other info you think is relevant, in particular if you have done any past research projects. There is a possibility of some financial support, so please indicate if you are interested in doing this as a URA for academic credit or for pay.

Readings

Some notes on the 2d simple random walk and Brownian motion.