Eventual coupling of planar symmetric random walks
= Eventual coupling of planar symmetric random walks
Two <simple symmetric random walks> on the planar integer lattice starting at sites of the same parity can be coupled to agree after a finite time almost surely. Run them independently until they meet, then give them identical increments. Their difference walk has return probability $p_{2n}(0,0)$ at time $n$; the divergent return sum proves <Markov-chain recurrence> and makes the meeting time finite.