Toth 2001 three favorite sites simple random walk

../

theorem_1: States that for simple symmetric random walk on the integers the expected number of steps onto a site that is one of exactly four favorite sites is finite, so that almost surely four or more favorites occur at only finitely many times.

---

Bálint Tóth, No more than three favorite sites for simple random walk, Ann. Probab. 29 (2001), no. 1, 484--503; DOI 10.1214/aop/1008956341 (Crossref record read (UTC)). Received February 2000; revised July 2000. AMS 2000 subject classifications 60J15, 60J55.

The copy read for this card is the publisher's typeset PDF of the twenty printed pages 484--503 (page head "The Annals of Probability 2001, Vol. 29, No. 1, 484--503"; physical PDF p. $n$ is printed p. $483+n$) with a clean text layer. The download's file name marked the copy as author-hosted, but the hosting URL was not recorded. Provenance: a survey download of September 2026; the download URL was not recorded; 142,876 bytes. No notice is printed in that copy; the publisher's article page gives the rights line "Copyright © 2001 Institute of Mathematical Statistics" (https://projecteuclid.org/journals/annals-of-probability/volume-29/issue-1/No-more-than-three-favorite-sites-for-simple-random-walk/10.1214/aop/1008956341.full, read 2026-10-02), every other right reserved.

Read status: claims checked. The definitions (1.1)--(1.9), the Question of Erdős and Révész, Theorem 1 and its two Remarks were read clause by clause in the text layer (pp. 484--486); the proof (sections 2--6, pp. 487--503) was not read beyond the overview on p. 486.

Contents

Setting (p. 484): $S_t$, $t\in\mathbb Z_+$, is simple symmetric random walk on $\mathbb Z$ with $S_0=0$; its local time is $L(t,x)=\#\{0<s\le t:S_s=x\}$ (1.3), the sum of the upcrossings $U(t,x)$ and downcrossings $D(t,x)$; the favorite (most visited) sites at time $t$ are $\mathcal K(t)=\{y:L(t,y)=\max_zL(t,z)\}$ (1.7). Formula (1.8) shows that at each step $\#\mathcal K$ is unchanged, increases by one (the current site becomes a new favorite), or drops to one (a favorite is revisited). The Question of Erdős and Révész (p. 485): does $\#\mathcal K(t)\ge r$ happen infinitely often, almost surely, for $r=3,4,\ldots$?

• Theorem 1 (p. 486; proof in sections 2--6): $\mathbb E\,f(4)<\infty$, where $f(r)=\#\{t\ge1:S_t\in\mathcal K(t),\ \#\mathcal K(t)=r\}$ (1.9) counts the steps onto one of exactly $r$ favorites, and $f(r+1)\le f(r)$. Remark 1: the negative answer for $r\ge4$ follows, that is, almost surely only finitely many times have four or more favorite sites (abstract and p. 485). Remark 2: the case $r=3$ remains open; the proof shows $\mathbb E\,f(3)=\infty$, and the author conjectures $f(3)<\infty$ almost surely.
• Overview of the proof (p. 486): $f(4)$ is written as a double sum of indicators of the moments when $\#\mathcal K$ rises from $3$ to $4$; the order of summation is inverted using the inverse local times (2.1)--(2.2); the Ray--Knight representation (section 3) turns the local time process at inverse local times into critical Galton--Watson processes; Proposition 1 (section 4, p. 491) bounds the relevant probabilities and expectations and is proved in section 5 from Lemmas 1--4 (among them a large-deviation estimate on the largest jump before hitting a level and the probability of hitting exactly a given level), with Side-lemmas 1--4 proved in section 6.
• Related results listed on p. 485: Bass and Griffin (transience of the favorite set), Csáki and Shi, Csáki, Révész and Shi (jumps of the favorite site), and the author's paper with Werner on favorite edges, whose starting ideas sections 1--3 follow.

Compiled scope

Only the introduction (pp. 484--486) was read. The proof was not checked, and nothing here is independently reviewed. The paper concerns the walk on $\mathbb Z$; it proves nothing about the planar walk of the problem.

Bears on. #1165, as the one-dimensional result to which the site attributes the bound for $r\ge4$; the planar answer compiled on that page is Hao, Li, Okada and Zheng, Theorem 1.1.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.