 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.