2D and 3D Walk: Drunk Man
A drunkard starts at the origin on an infinite lattice and at each tick moves one unit in a random coordinate direction (up/down/left/right in 2D; or one of six directions in 3D). For \(d=2\) and \(d=3\), what is the probability he ever returns to the origin, and what is the expected number of steps until his first return?
Answer
We have the following results:
2D: The random walk is null recurrent, i.e.:
\[ \Pr\{\text{return ever}\} = 1, \qquad \mathbb{E}[\text{first return}] = \infty. \]3D: The random walk is transient, i.e.:
\[ \Pr\{\text{return ever}\} \approx 0.3405 < 1, \qquad \mathbb{E}[\text{first return}] = \infty. \]