Continuity of percolation probability on hyperbolic graphs

LetT_k be a forwarding tree of degreek where each vertex other than the origin hask children and one parent and the origin hask children but no parent (k2). DefineG to be the graph obtained by adding toT_k nearest neighbor bonds connecting the vertices which are in the same generation.G is regarded as a discretization of the hyperbolic planeH² in the same sense thatZ^d is a discretization ofR^d. Independent percolation onG has been proved to have multiple phase transitions. We prove that the percolation probabilityO(p) is continuous on [0,1] as a function ofp.