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A circle packing is a collection of disks with disjoint interiors in theplane. It naturally defines a graph by tangency. It is shown that there exists$p>0$ such that the following holds for every circle packing: If each disk isretained with probability $p$ independently, then the probability that there isa path of retained disks connecting the origin to infinity is zero. Thefollowing conclusions are derived using results on circle packings of planargraphs: (i) Site percolation with parameter $p$ has no infinite connectedcomponent on recurrent simple plane triangulations, or on Benjamini--Schrammlimits of finite simple planar graphs. (ii) Site percolation with parameter$1-p$ has an infinite connected component on transient simple planetriangulations with bounded degree. These results lend support to recentconjectures of Benjamini. Extensions to graphs formed from the packing ofshapes other than disks, in the plane and in higher dimensions, are presented.Several conjectures and open questions are discussed.

On the site percolation threshold of circle packings and planar graphs