Do two drunk birds ever meet?
Theorem (Pólya, 1921). A simple random walk on \(\mathbb{Z}^d\) returns to its starting point with probability \(1\) if \(d \le 2\), but only with probability \(\approx 0.34\) if \(d = 3\).
So two walkers in space may never meet: their difference is again a random walk in \(\mathbb{Z}^3\). As Kakutani put it: “A drunk man will find his way home, but a drunk bird may get lost forever.”