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Claim. For discrete-time symmetric nearest-neighbor simple random walk on $\mathbb Z^2$ from the origin, with the time-zero visit counted, let $F(n)$ be the set of sites whose number of visits by time $n$ is maximal. Theorem 1.1 of C. Hao, X. Li, I. Okada and Y. Zheng, Favorite sites for simple random walk in two and more dimensions, Probability Theory and Related Fields 195 (2026), 1765–1822, recorded on its library card with the theorem's page, states that $\limsup_{n\to\infty}|F(n)|=3$ with probability one. Hence the probability Problem 1165 asks for is

$$\mathbb P\bigl(|F(n)|=r\text{ infinitely often}\bigr)= \begin{cases}1,&r=3,\\0,&r\ge4,\end{cases}$$

which answers the question for every integer $r\ge3$. The lower bound uses record levels of the maximum local time and two-point avoidance; the upper bound decomposes local times and screens candidate favorites in succession. The library's result pages reconstruct the theorem's proof from the 44-page arXiv version of 12 November 2025, with a weighted, parity-specific replacement for one printed conditional display, as the theorem's page explains; they do not cite the journal version's pagination.

Acceptance. Refereed: the paper appeared in Probability Theory and Related Fields, published online on 12 November 2025. Reviewed: Thomas Bloom, the curator of erdosproblems.com, labels the problem solved and credits this paper for the value $1$ at $r=3$; Bloom credits the value $0$ for $r\ge4$ to Tóth's 2001 paper, which concerns the walk on $\mathbb Z$, so the planar value for $r\ge4$ rests here on Theorem 1.1 as well, which gives it with the same limit superior. Tóth's result is a different dimension's and has no claim page. The page is dated by the first arXiv posting, 2 September 2024. A Lean formalization in Boris Alexeev's lean-proofs repository, linked above, names Hao, Li, Okada and Zheng as its informal authors and Codex and GPT-5.6 Sol as its formal authors, and states erdos_1165: for every $r\ge3$ the probability is $1$ if $r=3$ and $0$ otherwise. It has not been built or audited here, so no formalized evidence is listed.

Depends on. Nothing in this wiki: the argument is the paper's own.