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In this note we study some properties of infinite percolation clusters onnon-amenable graphs. In particular, we study the percolative properties of thecomplement of infinite percolation clusters. An approach based onmass-transport is adapted to show that for a large class of non-amenablegraphs, the graph obtained by removing each site contained in an infinitepercolation cluster has critical percolation threshold which can be arbitrarilyclose to the critical threshold for the original graph, almost surely, as papproaches p_c. Closely related is the self-destructive percolation process,introduced by J. van den Berg and R. Brouwer, for which we prove that aninfinite cluster emerges for any small reinforcement.

Bernoulli and self-destructive percolation on non-amenable graphs