Random Project

Consider the following two-dimensional random walk that is not confined to integer-valued coordinates: the walk starts at the origin, and each step is a randomly chosen unit vector. That is, to determine a step of the walk, first choose an angle θ (uniformly) from the interval [0, 2π). The step of the walk will then be (cos(θ),sin(θ)).

I will investigate the following questions:

1. What is the average squared distance from the origin after n steps?

2. When does the walk first return near the origin? Since the walk is not on a grid, it’s very unlikely that it will return exactly to (0, 0). Instead, when does it return to a small circle around the origin? For example, what is the average number of steps until the walk returns to a circle of radius 0.5 around the origin?

3. How does the walk behave if it is constrained to the region −5 ≤ y ≤ 5? Modify your code to keep the y-coordinate of the walk between −5 and 5. (You have freedom to decide how you would like to do this!) Then reconsider questions 1 and 2 for your modified random walk.