= Origin capture by a sufficiently sleepy planar random walk
Suppose two independent planar <simple symmetric random walks> are compared while one stays at its $j$th position throughout a deterministic time block. If each block has an unconditional probability bounded below by a positive constant that the other walk visits the origin, simultaneous origin visits occur infinitely often almost surely. <Markov-chain recurrence> gives infinitely many origin blocks for the sleepy walk. The block bound yields positive probability of infinitely many coincidences; <eventual coupling of planar symmetric random walks> then makes this future event have probability zero or one. Conditional block bounds or independence of capture attempts are not required.
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