CONTINUITY OF PERCOLATION PROBABILITY ON HYPERBOLIC GRAPHS

Let T-k be a forwarding tree of degree k where each vertex other than the origin has k children and one parent and the origin has k children but no parent (k greater than or equal to 2). Define G to be the grap h obtained by adding to T-k nearest neighbor bonds connecting the vert ices which are in the same generation. G is regarded as a discretizati on of the hyperbolic plane H-2 in the same sense that Z(d) is a discre tization of R-d. Independent percolation on G has been proved to have multiple phase transitions. We prove that the percolation probability theta(p) is continuous on [0,1] as a function of p.