Cover times for Brownian motion and random walks in two dimensions

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radius_distribution: Derives the exponential radius law with exact integer and boundary conventions.

theorem_1_1: The time simple random walk on the discrete torus (Z/nZ)^2 takes to visit every site, divided by (n log n)^2, tends to 4/pi in probability, proving Aldous's conjecture.

theorem_1_2: For Brownian motion on the two-dimensional torus, the time to come within epsilon of every point, divided by (log epsilon)^2, tends to 2/pi almost surely as epsilon tends to 0.

theorem_1_3: For Brownian motion on a smooth compact connected two-dimensional Riemannian manifold without boundary and of area A, the epsilon-covering time divided by (log epsilon)^2 tends to 2A/pi almost surely.

theorem_1_4: States the sharp distributional law for covering a disc centered at the origin.

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Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni, Cover times for Brownian motion and random walks in two dimensions, Annals of Mathematics 160 (2004), 433–464, DOI 10.4007/annals.2004.160.433.

Source versions

The copy read for this card is the published PDF, the 32-page publisher copy. Printed p. 433 is PDF p. 1. The publisher record identifies the volume, issue and page range.

An earlier copy read for this card is the 30-page arXiv v2. It identifies arXiv:math/0107191v2, 27 November 2003. Its PDF pagination is not the published pagination, and no line-by-line proof equivalence between the versions is asserted. All result citations below use the published version. The published PDF prints no notice on any of its 32 pages; the journal's article page, which links it as a free download, shows only the footer "Copyright © 2026 Annals of Mathematics" and names no license (https://annals.math.princeton.edu/2004/160-2/p02, read 2026-10-02), every other right reserved. For the arXiv copy, the arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0107191), every other right reserved.

Problem connection

Theorem 1.4, printed p. 436, settles the Kesten–Révész distributional conjecture for the time needed by a planar simple random walk to cover a disc centered at the origin. Its exact inversion gives the radius formulation of Problem 1164. The original 1999 question uses a random radius and an exponential cumulative distribution.

The radius deduction is fully written relative to the stated cover-time input. Theorem 1.4 itself remains a precise statement with a proof pointer: its complete same-paper chain has not yet been reconstructed here.

Main results

• Theorem 1.1, printed p. 434: the cover time $\mathcal T_n$ of the lattice torus $\mathbb Z^2/n\mathbb Z^2$ by simple random walk satisfies $\mathcal T_n/(n\log n)^2\to4/\pi$ in probability (Aldous's conjecture). Section 4, printed pp. 446–447, proves the lower half from Theorem 1.2 by strong approximation; the upper half is cited from Aldous and Fill.
• Theorem 1.2, printed p. 435: for Brownian motion on the flat torus $\mathbb T^2$, the $\varepsilon$-covering time satisfies $\mathcal C_\varepsilon/(\log\varepsilon)^2\to2/\pi$ almost surely as $\varepsilon\to0$. The upper bound is Section 2 (pp. 437–443), the lower bound Section 3 (pp. 443–446), with estimates from Sections 6 and 7 (pp. 452–458).
• Theorem 1.3, printed p. 435: on a smooth compact connected two-dimensional Riemannian manifold without boundary and of area $A$, the limit is $2A/\pi$ almost surely; Section 8, pp. 459–461.
• Theorem 1.4, printed p. 436: the disc-cover law above; Section 5, pp. 447–452.

The common method counts excursions between concentric circles on many scales at once and runs a second-moment argument for uncovered points while controlling the dependence on excursion endpoints. The torus walk of Theorem 1.1 is confined to $\mathbb Z_n^2$; Theorem 1.4 concerns the walk on all of $\mathbb Z^2$. Section 9 (pp. 461–462) adds Corollary 9.1 on the largest unvisited disc of the torus walk, recorded on the Theorem 1.1 page, and open problems.

Read status. Claims checked: Theorems 1.1–1.4 and the definitions they use (pp. 434–437, 447 and 459) were read clause by clause on the printed pages. The proofs were read for structure only; the radius deduction is this corpus's own argument from the stated Theorem 1.4. Nothing here is independently reviewed.

Bears on. #1164: Theorem 1.4 gives the limit law of the cover time of an origin-centered disc; the corpus's radius deduction inverts it to the law $\mathbb P((\log\max\{R_n,1\})^2/\log n\le x)\to1-e^{-4x}$, $x>0$, for the covered radius $R_n$, which implies the two-sided comparison $\log R_n\asymp\sqrt{\log n}$ in probability that the problem's corrected Statement asks about. Theorems 1.1–1.3 do not bear on the problem.

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