2-D random walks: simulation, video with R source code, curious facts

We have produced a 90-second video (click on this link to view the video) showing a'random walk' (a particular case of a Markov process) evolving over 400,000 steps. Figure 1 below shows the last frame (out of 2,000 frames, each one with 200 new steps). The video consists of 2000 frames, each showing 200 new steps (represented by red dots), as the random walk progresses over time. Sometimes we seem to get stuck locally, sometimes we are dramatically progressing at high speed. Older frames still stay on the video, but are shown in gray rather than red.