Hao 2024 favorite sites simple random walk two

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domino_pairing_transfer: Supplies the pairing-invariant holding-time argument omitted from the source's treatment of the two striped pairings in Proposition 4.7.

favorite_union_corollary: Proves the logarithm-squared bound on the union of planar favorite sets and the upper limit constant three over pi.

lemma_2_1: Records the planar return, two-point avoidance, and escape estimates imported by the favorite-site proof, with small-distance conventions.

lemma_2_2: Bounds the probability that a transient simple random walk hits any of k specified sites after time n by a constant times k n to the 1-d/2.

lemma_2_3: Gives the negative-binomial local expansion with its missing square-root normalization restored and identifies why fixed-index ratios are unchanged.

lemma_2_4: Derives both moderate-deviation tails for geometric holding times from the exact moment generating function and exponential tilting.

lemma_2_5: Proves the local-time tail bounds for the original and shortened planar walks using return generating functions and the local central limit theorem.

lemma_2_6: Deduces an exponentially small tail for the creation time of k tied favorites from the planar maximum-local-time deviation estimate.

lemma_2_7: Proves an exponential bound for many balls occupying the highest occupied urn when the next urn has a comparable sampling probability.

lemma_2_8: Bounds extra balls near the highest occupied urn by the relative width of that band and the number of balls in a wider band.

lemma_3_1: Defines the stopped avoidance events precisely and proves that their late components eventually characterize simultaneous favorites.

lemma_3_2: Proves a polynomial all-future bound for maximum-local-time regularity and the absence of short path segments joining two thick sites.

lemma_3_3: Uses thick-site separation to exclude early visits to previous favorites and to bound the time between successive new favorites at one level.

lemma_4_1: Proves a uniform inverse-square-root lower bound for creating another planar favorite while avoiding the one or two existing favorites.

lemma_4_10: Shows that a new nearby favorite is very unlikely to start far below the record level, with the candidate set corrected to omit old favorites.

lemma_4_11: Proves the geometric growth bound for every local-time slice, including the separate first-slice likelihood comparison.

lemma_4_12: Proves that negative-binomial probabilities remain comparable across the short local-time intervals used in the screening argument.

lemma_a_2: Proves the required conditional excursion comparison from precise annulus and Harnack inputs, including the logarithmic error factor.

lemma_a_4: Gives the uniform first-moment estimate for successful excursion profiles, with the exponent required by the later quantitative argument.

lemma_a_6: Derives the negative-binomial count model and the profile comparisons needed for the first and second moments, including omitted count ranges.

lemma_a_8: Bounds the discrete Gaussian path sum by Brownian confinement, with integer endpoints and a proved small-ball estimate.

local_time_decomposition: Defines the two domino-based local-time decompositions with explicit endpoint corrections and repairs the printed primed parity convention.

proposition_1_3: Proves the stretched double-exponential lower-deviation bound through the complete Appendix A excursion and moment argument.

proposition_4_2: Proves conditional independence and the negative-binomial laws of erased excursions, including the terminal-visit correction.

proposition_4_3: Proves the independent truncated negative-binomial laws on dominoes containing no designated favorite at a record stopping time.

proposition_4_4: Bounds the number of large shortened-walk local times at the deterministic time used to screen possible planar favorite sites.

proposition_4_5: Proves that a near-favorite site's external local time is close to fifteen-sixteenths of its total, outside a stretched-exponential event.

proposition_4_7: Proves the pairing-dependent four-favorite bound by combining the two screening scales with planar escape estimates.

proposition_4_8: Bounds the number of near-favorite sites throughout the wide local-time range used in the planar four-favorite argument.

proposition_4_9: Proves the parity-specific screening bounds and the weighted form needed in the four-favorite reduction, while isolating a conditioning gap in print.

proposition_a_3: Proves the excursion-profile moment bounds and the disk-exit lower tail, including local-time concentration and the separated-point argument.

proposition_a_7: Proves the sharp exponential cost of the excursion-count profile, including its Gaussian comparison and integer block decomposition.

record_levels: Relates simultaneous favorites to stopping times at successive local-time levels and gives the filtration needed for conditional Borel–Cantelli.

theorem_1_1: States the almost-sure planar favorite-count limit and reconstructs its complete proof from the record, creation-time, and screening estimates.

theorem_1_2: Proves the sharp iterated-logarithm favorite-count limit in transient dimensions using late avoidance and ordinary and conditional Borel–Cantelli.

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Chenxu Hao, Xinyi Li, Izumi Okada, and Yushu Zheng, Favorite sites for simple random walk in two and more dimensions. The copy read for this card is arXiv:2409.00995v2, dated 12 November 2025, 44 pages.

The paper was published online on 12 November 2025 in Probability Theory and Related Fields, volume 195, pp. 1765–1822 (2026), DOI 10.1007/s00440-025-01441-1. The complete 58-page journal PDF has not been acquired. All result labels and page references on this card and its result pages are to the 44-page arXiv v2. Publication establishes acceptance of the work; it does not establish identity of the unavailable published proof and this preprint. The 46-page arXiv v1 is dated 2 September 2024. A comparison of labeled statements is not a full version audit, and no equivalence of the complete proofs across editions is asserted. The arXiv record names arXiv's non-exclusive distribution license for both versions (arXiv:2409.00995), every other right reserved.

For discrete-time symmetric nearest-neighbor simple random walk on $\mathbb Z^d$, with time zero counted in local times, a favorite site is one attaining the current maximum local time. Theorem 1.1 proves that the planar favorite count has almost-sure limit superior three: three favorites occur infinitely often and four or more occur only finitely often. This gives the probability values in Problem 1165. The favorite-union corollary combines its eventual upper bound with Erdős–Taylor's maximum-local-time bound to settle Problem 1166, with union size $O((\log n)^2)$ and upper limit constant at most $3/\pi$. The latter is an upper bound, not a sharpness assertion for the union.

The Erdős–Révész questions appear as Problem 6.77 and Problem 6.78 in Some of Paul's favorite problems, the July 1999 conference booklet compiled from various contributors, cited as [Va99] on the problem site. The paper (p. 2) traces the favorite-count question to Erdős and Révész (1984) and (1987), its references [14] and [15].

In dimensions $d\ge3$, Theorem 1.2 gives the sharp almost-sure limit superior $\limsup_{n\to\infty}|K^{(d)}(n)|/\log\log n=-1/\log\gamma_d$, where $\gamma_d$ is the escape probability. It refines the earlier Erdős–Révész result that every fixed favorite count occurs infinitely often in those dimensions. The one-dimensional results of Tóth (2001) and Ding–Shen (2018) concern a different dimension. In particular, the E1165 site snapshot's attribution of the planar upper bound to Tóth is not supported by Tóth's paper; both planar bounds are supplied by the present Theorem 1.1.

The planar proof measures time by maximum-local-time levels. The record-level identities and Lemma 4.1 give the lower bound using two-point avoidance and conditional Borel–Cantelli. The upper bound decomposes local time into erased backtracking excursions and the remaining path. It then screens near-favorite candidates before applying a summable four-favorite bound. Proposition 1.3 is the quantitative maximum-local-time lower-deviation estimate needed for that screening, proved by an adaptation of Rosen's excursion method.

Proof coverage. The full planar Theorem 1.1 argument is reconstructed, including the record-level lower bound, the local-time decompositions, the candidate-screening estimates, the transfer to the required domino pairings, and the final summability argument. Its quantitative input Proposition 1.3 and the six supporting Appendix A pages have complete rewritten proofs. They cover excursion decoupling, upcrossing counts, first and second moments, Gaussian confinement, and amplification to deterministic time. The exact domains of their event and parameter restrictions remain in force.

There is a specific qualification to this coverage. The stronger printed conditional display (4.35) in Proposition 4.9 remains uncertified: it conditions a union of both parity classes on one shortened path, although the two conditional product laws use different paths. The reconstruction proves separate parity-specific estimates and integrates them against weights determined by the favorite locations. This gives precisely the estimate used in Proposition 4.7. Theorem 1.1 and the favorite-union corollary use that proved replacement, not the stronger printed display. No assertion that (4.35) is false is made.

The transient argument is also written in full: Theorem 1.2 and Lemmas 3.1–3.3, together with the specialized two-site occupation law of Csáki–Földes–Révész–Rosen–Shi and the transient maximum-local-time theorem of Erdős–Taylor, form six full deductions. The Csáki input page proves the simple-walk specialization needed here, not the general finite-set spectral theorem. The standard heat-kernel, local central limit, Green-function, annulus, and conditional Borel–Cantelli inputs remain explicitly stated external results; the Appendix A pages identify their precise additional Rosen input. Full reconstruction of those external works or every ancillary claim in the source is not asserted.

Source corrections. The corrected prefactor and its derivation are recorded in Lemma 2.3. Equation (3.14) on p. 13 needs an intersection in place of its printed union, as explained on the Theorem 1.2 page. The p. 6 primed lazy-time notation also has a parity inconsistency: the erased odd excursion starts at the odd member of a domino, whereas the displayed counting formula uses its even member. The decomposition proof assigns that count and its half-excursion to the correct endpoints. The Appendix A and screening pages likewise state their local corrections and the deductions replacing them. These are compilation-supplied repairs, not author-issued errata or statements about the unavailable journal edition.

Literature check, 2026-09-05. The arXiv history, publisher record, Hao's institutional page, author-page searches, and web searches for later papers and indexed X announcements were checked. No later planar replacement or materially distinct accepted proof was located in this bounded search. Later work on asymmetric one-dimensional walks, rarely visited sites, and favorite edges has a different object or walk law. This is not an exhaustive absence claim. That search missed a Lean formalization of the answer to Problem 1165, Erdos1165.lean in Boris Alexeev's lean-proofs repository, added on 2026-08-22. It names this paper's authors as its informal authors and Codex and GPT-5.6 Sol as its formal authors, and the Lean proof linked for Problem 1166 imports it. It was not built or audited here.

Bears on. #1165 and #1166.

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