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