The Drunkard and the Bird

The problem
random walk on a latticeorigindoes the walker eventually return to start?

A walker stands at the origin of an infinite lattice. Every second, they take one step in a uniformly random direction along the lattice axes.

  • 1D: left or right.
  • 2D: up, down, left, or right.
  • 3D: six choices (add forward, back).

Does the walker eventually return to the origin? Does the answer depend on dimension?