Continuity of percolation probability on hyperbolic graphs
LetTk 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 (k≥2). DefineG to be the graph obtained by adding toTk 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 thatZd is a discretization ofRd. 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.
Key WordsPercolation percolation probability hyperbolic graphs
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