## Monday, March 30, 2015

### Intuitive Proofs

As I mentioned a few months ago, I briefly joined an undergraduate research seminar my freshman year at Cornell. In that seminar I was asked if a two-dimensional random walk on a lattice would return to the origin infinitely often. I said of course. The advisor was impressed until he asked about three-dimensional walks and I said they also hit the origin infinitely often. My intuition was wrong.

33 years later I'd like to give the right intuition. This is rough intuition, not a proof, and I'm sure none of this is original with me.

In a 1-dimensional random walk, you will be at the origin on the nth step with probability about 1/n0.5. Since the sum of 1/n0.5 diverges this happens infinitely often.

In a 2-dimensional random walk, you will be at the origin on the nth step with probability about (1/n0.5)2 = 1/n. Since the sum of 1/n diverges this happens infinitely often.

In a 3-dimensional random walk, you will be at the origin on the nth step with probability about (1/n0.5)3 = 1/n1.5. Since the sum of 1/n1.5 converges this happens finitely often.