This animation shows a random walk in the hyperbolic disk with step size 0.02. In other words, take a step of (hyperbolic) length 0.02 straight ahead, then turn in a random direction, and repeat. At this scale it is a good approximation of Brownian motion. Furstenburg proved that Brownian motion in the hyperbolic plane almost always converges to a single point at infinity. This can be observed here, though it can take a while. The random walks shown take about 50000 steps before restarting. Be patient, the script can take a while to load. The random walk will repeat itself, and the animation is smoother after the first run.

Change the random walk. Return to the home page.

These animations were made in 2008 by Peter Storm using Mathematica and JavaScript.