Pólya's theorem

Simple random walk on the lattice $\mathbb Z^d$ moves from a point to one of its $2d$ nearest neighbours (one coordinate changed by $\pm1$) uniformly at random. The walk is recurrent when return to the starting point is certain — the probability of no return by time $t$, $\mathbb P_0\{\tau^+_0>t\}$, tends to $0$ — and transient when with positive probability it never returns.

The theorem (Pólya's theorem; §21.2, Examples 21.8–21.9 of Levin–Peres–Wilmer — the capstone of Chapters 20–21) asserts:

1. in dimensions $d=1$ and $d=2$, the walk is recurrent;
2. in every dimension $d\ge3$, the walk is transient.

"A drunk man will find his way home, but a drunk bird may get lost forever." The dichotomy is decided by the Green's-function criterion of this mission: the return probabilities obey $P^{2t}(0,0)\asymp t^{-d/2}$ (a local central-limit estimate, by Stirling's formula in low dimension), and $\sum_tt^{-d/2}$ diverges exactly for $d\le2$. Pólya's 1921 theorem inaugurated the dimension-dependent study of random walks; formalizing the transient half in particular requires genuinely quantitative control of $d$-dimensional return probabilities.