Random walks on hyperbolic space

We show the distribution of a random walk for the SL(2,R) action on hyperbolic space (in this case arising as the SU(1,1) acting on 0 in the hyperbolic disc). We chose a distribution on SL(2,R) supported on two elements. The images below show the distribution of the associated random walk after n steps, where n ranges from 1 to 15.