Erdős Problem #1165

Given a random walk $s_0,s_1,\ldots$ in $\mathbb{Z}^2$, starting at the origin, let $f_n(x)$ count the number of $0\leq k\leq n$ such that $s_k=x$.

Let\[F(n)=\{ x: f_n(x) = \max_y f_n(y)\}\]be the set of 'favourite values'. Find\[\mathbb{P}(\lvert F(n)\rvert=r\textrm{ infinitely often})\]for $r\geq 3$.

#1165: [Va99,6.77]

probability

A problem of Erdős and Révész.

Tóth [To01] proved that this probability is $0$ for all $r\geq 4$. Hao, Li, Okada, and Zheng [HLOZ24] proved that this probability is $1$ for $r=3$.

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This page was last edited 27 January 2026.

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Additional thanks to: Sarosh Adenwalla

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