  The percolation transitions on hyperbolic lattices are investigated
numerically using finite-size scaling methods. The existence of two distinct
percolation thresholds is verified. At the lower threshold, an unbounded
cluster appears and reaches from the middle to the boundary. This transition is
of the same type and has the same finite-size scaling properties as the
corresponding transition for the Cayley tree. At the upper threshold, on the
other hand, a single unbounded cluster forms which overwhelms all the others
and occupies a finite fraction of the volume as well as of the boundary
connections. The finite-size scaling properties for this upper threshold are
different from those of the Cayley tree and two of the critical exponents are
obtained. The results suggest that the percolation transition for the
hyperbolic lattices forms a universality class of its own.

Statistical Mechanics