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a.s. convergence atom at zero Bernoulli distribution Bernoulli percolation bond percolation Borel-Cantelli lemma bution closed bonds closed cluster consider converges a.s. coordinate distribution Corollary covariant critical probability crosses cut set cylinder first-passage cylinder path defined Definition denote the number distribution with U(0 distributive lattice dual graph dual lattice event exponential distribution finite first-passage percolation first-passage processes fixed fluid follows given Hammersley and Welsh height process Hence implies independent infinite cluster infinite open cluster intersection Lemma lim inf lim sup monotone non-negative number of bonds open bonds open circuit open cluster containing open path origin path from 0,0 percolation model percolation problem percolation processes positive integer proof of Theorem proved random variables reach process route length self-avoiding paths Seymour and Welsh site percolation sponge square lattice subadditive function subadditive process Theorem 5.1 uniformly integrable upper bound vertices x-axis