Lost in three dimensions, chance of ever returning

Question

A walker starts at the origin of an infinite 3D lattice. Each step it moves one unit along one of the six axis directions (up, down, north, south, east, west), each with probability $1/6$ , and it walks forever.

Roughly what is the probability the walker eventually returns to its starting point?

Accepted Answer

In three dimensions there is finally enough room to escape, so the symmetric lattice walk is transient: it may drift off and never come back. The exact return probability is a famous constant computed by Polya, P(	ext{eventual return}) \approx 0.3405, about 34%. So in 3D the walker is more likely than not to never return, a sharp contrast with the certain return in 1D and 2D. Takeaway. The 3D symmetric walk is transient with return probability near 0.34; adding a dimension turns a sure return into a coin-flip-and-worse.

Lost in three dimensions, chance of ever returning

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