A drunk person wandering a city grid will always find their way home.
But in our 3D universe? They're lost forever.
In one or two dimensions, a random walker returns to the origin
with probability 1 — certainty. They will always come back.
In three dimensions, the probability drops to just
≈ 34%. Add more dimensions, and they're almost certainly
lost forever.
In lower dimensions, there's simply less space to get lost in. A drunk on a 1D line can only go left or right — eventually, they'll stumble back to the starting point.
In 2D, despite having more freedom, the walker's path is still "squeezed" enough that return is inevitable. They might wander far, but the probability of returning remains 1.
In 3D, the vast emptiness of space opens up. The walker has so many directions to choose from that they can drift away infinitely without ever finding their way back.
Pólya proved in 1921 that recurrence depends on whether a certain infinite series converges or diverges.
For d=1,2: the series diverges → guaranteed return
For d≥3: the series converges → finite probability of never returning
The value u₃ is related to Watson's triple integral, a famous result in mathematical physics connected to random walks on lattices.