Hyperbolic Random Walk | hypertiling

Three groups of walkers compete to paint a hyperbolic tiling in each group's color. In every time step, the walkers move to one of three adjacent cells, leaving the cell behind dyed in their color. In a hyperbolic space, the phenomenology of a random walk and its continuous-time counterpart, the Brownian motion, is drastically different from that on the flat Euclidean space. Due the space being more expansive/spacious in every direction, motion becomes ballistic on scales larger than the curvature radius, i.e. the mean distance traveled scales linearly with time, instead of with the square root of time, as is the case for ordinary diffusion. Moreover, much like as in Euclidean spaces of dimension three or higher, Brownian motion in hyperbolic spaces is not recurrent, i.e. the probability of a symmetric random process to return to the origin vanishes as the lattice size increases.

The lattice is a regular hyperbolic (3,7) tiling constructed with our hypertiling Python package. For a better resolution we applied two levels of triangle refinements, therefore the final tessellation consists of 21.136 triangular cells.

#hyperbolicspace
#poincaredisk
#randomwalk

Further reading:

- Grigor'yan, A., & Noguchi, M. (1998). The heat kernel on hyperbolic space. Bulletin of the London Mathematical Society, 30(6), 643-650.
- Monthus, C., & Texier, C. (1996). Random walk on the Bethe lattice and hyperbolic Brownian motion. Journal of Physics A: Mathematical and General, 29(10), 2399.
- Kendall, W. S. (1984). Brownian motion on a surface of negative curvature. Séminaire de probabilités de Strasbourg, 18, 70-76.
- Sausset, F., & Tarjus, G. (2008). Self-diffusion in a monatomic glass-forming liquid embedded in the hyperbolic plane. Philosophical Magazine, 88(33-35), 4025-4031.