Independent annular barriers for percolation
= Independent annular barriers for percolation
Closed dual <graph cycles> in disjoint square rings depend on disjoint primal <edge> sets. In independent <bond percolation> their barrier events are independent. If each event has <probability> at least $\delta>0$, an open <graph path> from the origin across $k$ rings has <probability> at most $(1-\delta)^k$. A geometric sequence of radii turns this estimate into the <polynomial critical one-arm upper bound>.