Erdős Problem #1164

Let $R_n$ be the maximal integer such that almost every random walk from the origin in $\mathbb{Z}^2$ visits every $x\in\mathbb{Z}^2$ with $\| x\|\leq R_n$ in at most $n$ steps.

Is it true that\[\log R_n \asymp \sqrt{\log n}?\]

#1164: [Va99,6.76]

probability

A problem of Erdős and Taylor. This is true, as proved independently by Révész [Re90] and Kesten.

Both made the stronger conjecture that\[\lim_{n\to \infty}\mathbb{P}\left(\frac{(\log R_n)^2}{\log n}\leq x\right)=e^{-4x}\]for all $x>0$. This was proved by Dembo, Peres, Rosen, and Zeitouni [DPRZ04].

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This page was last edited 25 January 2026.

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Additional thanks to: Alfaiz and Terence Tao

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