Random walk on the Bethe lattice and diffusion on the hyperbolic plane
The diffusion on the hyperbolic plane (Poincaré half plane), a surface of constant negative curvature, and the random walk on the Bethe lattice, present some similarities. In both cases, the diffusion in the radial coordinate can be mapped onto a one dimensional diffusion with a drift. For the Poincaré half plane the drift originates from the negative curvature, whereas in the case of the Bethe lattice it is generated by the exponential growth of the number of sites.
A close relation between the two problems is expected by noticing that Bethe lattice is a regular tiling of the hyperbolic plane. Note however that the definition of a continuum limit of a random walk on tessellations (tiling) (1) (p,q) of the hyperbolic plane is made more complicated by the fact that the lattice spacing is a function of the parameters (p,q).
(1) (p,q) designates a tessellation by p-gones with coordinence q. Tessellations of the hyperbolic plane satisfy 1/p+1/q<1/2 (on a flat surface, the three possible tessellations (3,6), (4,4) and (6,3) satisfy 1/p+1/q=1/2).