Summary: Here is a set lectures by Sid Redner. There exists several simple models of random walks. It can be either fixed stepsize but random directions or varying stepsize. Fixed Stepsize A simple model is that we assume each step is a step of size $ \epsilon$, but with random directions. The position of this random walker at step $ N$ is the summation of all the steps (vectors), $$ \vec X = \sum_i^N x_i, $$ where $ x_i$ is the vector that represents step i.