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 ≥ 2). Define G to be the graph obtained by adding to T_k nearest neighbor bonds connecting the vertices which are in the same generation. G is regarded as a discretization of the hyperbolic plane H² in the same sense that Z^d is a discretization of R^d. Independent percolation on G has been proved to have multiple phase transitions. We prove that the percolation probability 0(p) is continuous on [0,1] as a function of p.