1 Pólya’s theorem

The goal of this project is to prove that:

A drunk man will find his way home, but a drunk bird may get lost forever.

— Shizuo Kakutani

https://mathshistory.st-andrews.ac.uk/Biographies/Kakutani/quotations/

Somewhat more mathematically, the goal is the following:

Theorem 1.1 Pólya’s theorem

The simple random walk \(X= \big( X(t) \big)_{t \in \mathbb {N}}\) on the \(d\)-dimensional grid \(\mathbb {Z}^d\) is recurrent if \(d \le 2\) and transient if \(d \, {\gt} \, 2\).

Proof ▶

Theorem 1.2 below asserts that \(X\) is expectation recurrent iff \(d \le 2\), and Lemma 3.4 shows that recurrence and expectation recurrence are equivalent for the simple random walk.

The essence of the proof is to establish the following slightly modified version of the theorem.

Theorem 1.2 Pólya’s theorem, alternative form

The simple random walk \(X= \big( X(t) \big)_{t \in \mathbb {N}}\) on the \(d\)-dimensional grid \(\mathbb {Z}^d\) is expectation recurrent if \(d \le 2\) and expectation transient if \(d \, {\gt} \, 2\).

Proof ▶

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