Why a Drunk Man Can Always Return Home and a Bird Can’t

I bet you can picture this situation: you are drunk, you left your college dorm and now you are randomly walking with no idea of where your dorm is. Above you a beautiful bird is also flying trying to find his nest. Mathematics demonstrates that you can return to your house, but the bird can’t.

The walk of a drunk person can be modelled by what is called a Markov chain*. Each step you take, you have a certain probability to go left, right, ahead or behind. And every time you take one step, it does not depend on the previous steps. We modelize the coordinates of our person by an element of ℤᵈ, where d here is either 2 (it is us, we walk on a 2D plane) or 3 (the bird can fly up and down, too, in a 3D space).

What we want is to determine when our Markov chain goes through the starting point. We are going to write Xₙ, the random variable who will represent our coordinates on the nth step. A very important theorem, named Polya’s theorem, assures us that

Theorem 1 Σℙ(Xₙ = 0) diverges ⇔ Xₙ goes through its starting point an infinite number of times

With some combinatorics and analytics tricks, we can prove that the series diverge if and only if d=1 or d=2**.

Therefore, it means that our bird, who can fly in a three-dimensional space, only goes back to its starting point in specific cases. Additionally, the higher d is, the smaller is the probability for the drunk guy, or the bird, to go back home. So the next time you wish people could have wings, you better think twice: if we could fly, we wouldn’t go back home when we are drunk!

*A Markov chain is a mathematical system that undergoes transitions from one state to another within a finite or countable number of possible states. The key characteristic of a Markov chain is that the probability of transitioning to any particular state depends only on the current state and not on the sequence of states that preceded it

**The variable d refers to the dimension of space in which the random walk occurs

• d=1: the walk is in a one-dimensional line

• d=2: the walk is in a two-dimensional plane

• d=3: the walk is in three-dimensional space