{"id":"c94276ef-2276-4e86-8589-349a4f052520","arxiv_id":"2411.16398","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For d ≥ 3 and γ > (d+2)/(2d), the probability that the torus cover time is at most γ times its mean is exp(-(1+o(1)) N^{d(1-γ)}).","lead":"This paper proves sharp large-deviation asymptotics for the probability that a random walk on a d-dimensional torus covers all sites faster than its expected time, confirming earlier conjectured exponents for d ≥ 3. The proof introduces a new loop-surgery technique and uses a recent strong coupling between the walk and random interlacements.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The lower bound for Theorem 0.1 rests on Lemma 1.7, an unproved generalization of [26, Lemma 6.1] to the N-dependent late-point regime; if the asserted uniformity fails, the concentration of late points (Lemma 1.8/3.4) and hence Stages 1–3 of the proof lose their foundation.","rationale":"Theorem 0.1 is carefully structured: the upper bound is a standard cube argument using the explicit coupling error, and the lower bound's four-stage construction is internally consistent. The threshold γ>(d+2)/(2d) is transparently the point at which the coupling error exp(-cN^{(d-2)/2}/log N) is negligible against exp(-N^{d(1-γ)}). The loop-insertion surgery (Section 4) is detailed and the recovery argument for (4.1c) is plausible; no concrete error found there. The one step that is simultaneously essential and unsupported is Lemma 1.7. The paper explicitly delegates its proof to 'the same proof as [26, Lemma 6.1]' and says the generalization 'still goes through,' but does not show the reader how the N-dependent α_N and the summed two-point probabilities interact with the error term. Because Lemma 1.8, Lemma 3.4, Proposition 3.5, and Proposition 3.7 all consume estimates derived from Lemma 1.7, this is the hinge on which the lower bound turns. This is a valid reason to keep the verdict CONDITIONAL rather than ACCEPT: the central idea is sound and the gaps are plausibly fillable, but a full verification of Lemma 1.7 (or a replacement proof) is needed before every claim in the paper is established. The reader's weakest_assumption named exactly this unproved generalization, so I agree with that identification. My recommendation is UNCHANGED: the reader's CONDITIONAL verdict already reflects this concern and no new objection beyond it has surfaced.","tokens_in":37253,"tokens_out":24997,"duration_ms":216599,"concrete_test":"Locate [26, Lemma 6.1] and its proof. Verify (i) the stated range of α: if it is already 'uniform for α in any compact subinterval of (0,1)', then check that the error constant does not depend on how α_N approaches γ; (ii) the class of G: the proof must give the bound for two-point sets G={x,y} of bounded capacity uniformly over all pairs on the torus; (iii) plug α_N = γ - K/(d g(0) log N) into the proof and track all terms of order (α_N-γ) or 1/log N to confirm they are absorbed in the stated (log N)^{3/2}/N^{(d-2)/2} error. If (i)-(iii) hold, the concern is resolved; if the original lemma has an extra restriction or a hidden dependence on the rate of approach, Lemma 1.7 must be proved independently and the prefactor in Theorem 0.1 re-checked.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 1.7 is the quantitative basis for the structure of late points used throughout the lower bound. It asserts P(G⊂L_α)/(N^{-dαg(0)cap(G)}) - 1 = O((log N)^{3/2}/N^{(d-2)/2}) uniformly in α∈[α0,α1] and G with cap(G)≤β0. The proof is not given: the text says '[26, Lemma 6.1] ... still goes through under such generalization.' But the application needs more than the original statement if the original was only for fixed α in a compact subinterval: here α_N = γ - K/(d g(0) log N) varies with N and approaches γ∈(γ0,1), and Lemma 1.9 sums the two-point version over ~N^{2d} pairs. If the uniformity in α is not actually available at the stated rate, the o(1) in Lemma 1.9 could acquire a logarithmic or worse factor. Since Lemma 1.8 (concentration of |L_α^F|) is derived from Lemma 1.9, and Lemma 3.4 (E1 regularity and separation) is derived from Lemma 1.8, and Propositions 3.5 and 3.7 (Stages 2 and 3) use the same regularity bounds, a failure of Lemma 1.7 would invalidate the lower-bound event E and the exponential estimate in Proposition 3.1. This is the weakest link in the chain from the coupling to the sharp prefactor -1. The gap is openly acknowledged by the authors, but it is load-bearing, not cosmetic. Other supporting statements (Theorem 0.2's Lemma 3.8, Appendix B) are also asserted without proof, but they do not feed into Theorem 0.1; Lemma 1.7 does.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies downward large deviations of the cover time C_N of the discrete torus (Z/NZ)^d, d ≥ 3, by simple random walk. The main result, Theorem 0.1, states that for γ in ((d+2)/(2d), 1), the probability that C_N ≤ γ t_cov, with t_cov = g(0)N^d log N^d, satisfies log P(U_{γ,N}) / N^{d(1-γ)} → -1. The proof combines the recently developed strong coupling between the random walk trace and random interlacements (Prévost–Rodriguez–Sousi) with a four-stage lower-bound construction: late-point regularity, bulk coverage via interlacements, explicit loop-insertion surgeries for the edge, and final free roaming. The paper also states Theorem 0.2, a lower bound of the form exp(-C N^{d(1-γ)}) for all γ ∈ (0,1), and Corollary 0.3, which upgrades this to exp(-N^{d(1-γ)+o(1)}) using a sketched upper bound (0.6). Appendix B sketches a related sharp asymptotic for the cover level of random interlacements on cubes.","tokens_in":37679,"tokens_out":8833,"duration_ms":80751,"significance":"If the proof is completed, Theorem 0.1 would be the first sharp large-deviation prefactor for the cover time of tori in dimensions d ≥ 3, matching the conjecturally correct exponent from the upper bounds of Goodman–den Hollander and Comets–Gallesco–Popov–Vachkovskaia. The paper articulates clearly why the threshold γ > (d+2)/(2d) arises from the coupling error in Proposition 1.5, and it honestly notes that an improved coupling would extend the range. The deterministic loop-insertion surgery in Section 4 is an interesting and potentially reusable construction, and the upper-bound argument in Section 2 is concrete, with explicit error terms. The main weakness is that a load-bearing technical lemma on late points, Lemma 1.7, is asserted as a generalization of a result from [26] without proof; several auxiliary statements are also only sketched. These gaps are acknowledged in the text but are not merely cosmetic, because the lower bound of Theorem 0.1 depends on the uniformity in α stated in Lemma 1.7.","major_comments":[{"comment":"Lemma 1.7 is stated as a uniform generalization of [26, Lemma 6.1] to all α ∈ [α0, α1] with error O((log N)^{3/2}/N^{(d-2)/2}), but no proof is given; the text only says that the proof 'still goes through.' This is load-bearing for Theorem 0.1: Lemma 1.9 uses (1.17) to sum two-point probabilities over ~N^{2d} pairs, Lemma 1.8 derives the concentration of |L_α^F| from Lemma 1.9, and Lemma 3.4 together with Propositions 3.5 and 3.7 use that concentration to control the event E1 and hence the event E in Proposition 3.1. Since α_N = γ - K/(d g(0) log N) varies with N and approaches γ, the uniformity in α at the stated rate is exactly what the lower bound needs; a weaker uniformity could introduce logarithmic factors into (1.20), destroying the o(1) in Lemma 1.9 and the concentration estimate (1.19). Please provide a complete proof of the uniform version of Lemma 1.7, or state the precise hypotheses that the application requires and verify them.","section":"§1.3, Lemma 1.7 and Lemma 1.9"},{"comment":"The proof of Proposition 3.1 invokes 'the weaker version of Proposition 3.6 with d∞(x, X[0,4εN^d - 1]) in place of r^{3ε}_x,' but Proposition 3.6 and its proof are written only for the 3ε time scale, while the event E3 in (3.10) concerns the interval (T2, T3] of length 4εN^d. Since Proposition 3.1 is the key exponential estimate for the lower bound in Theorem 0.1, the version actually used must be stated and proved. Please state the 4ε analogue of Proposition 3.6 and give the short adaptation of the proof of Proposition 3.7, or adjust the parameters so that the statement matches the proof.","section":"§3.2, proof of Proposition 3.1"},{"comment":"Lemma 3.8, which asserts the existence of a high-probability event E′ with the required regularity and distance bounds, is stated with 'We omit the proof.' Theorem 0.2 rests entirely on this lemma together with Proposition 3.3, so the proof cannot be omitted from a formal paper. Please include a full proof, or a precise reduction to Lemmas 1.8 and 1.9 with all constants and uniformity statements tracked.","section":"§3.3, Lemma 3.8 and Theorem 0.2"}],"minor_comments":[{"comment":"The definition K_N = Q(0, N^{1+δ}) ∩ (s_N Z)^d appears to be a typo: with this cube, the claimed properties that K_N is (1+δ)R_N-well separated and that Q(x, R_N) ⊂ Q_{δ/2} for all x ∈ K_N fail, and the count |K_N| ∼ (N/(1+δ)/s_N)^d in (2.4) corresponds to a cube of side length N/(1+δ). Please correct this to Q(0, N/(1+δ)) or the intended analogue, both in Section 2 and in Appendix B.","section":"§2 and Appendix B, definition of K_N"},{"comment":"The phrase 'self-disjoint loop' in Table 5.3 should be 'self-avoiding loop', which is the term used elsewhere in Section 4.","section":"§4.1 and Table 5.3"},{"comment":"The same symbol P (and E) is used both for the law of the random walk on the torus and for the law of random interlacements; this is a recurring source of potential confusion, especially in Propositions 1.4 and 1.5 where extended measures are denoted eP. Consider using a different symbol such as P^I for the interlacement law.","section":"§1.1 and throughout"},{"comment":"There are several typos in the exposition, for example 'we devide' in the description of Stage 3 and 'sa espérance' in the French résumé; these should be corrected in a final revision.","section":"§0.1"}],"recommendation":"major_revision","confidential_remarks":"The authors are transparent about the main gap: Lemma 1.7 is an unproved uniform generalization of a lemma from a recent preprint, and the lower-bound proof of Theorem 0.1 depends on it. I do not see circularity or parameter fitting, and the overall strategy is credible. The missing proofs of Lemma 1.7 and Lemma 3.8 are likely fillable, but they are load-bearing, so the manuscript should not be accepted until they are supplied. The repeated Q(0, N^{1+δ}) typo in Section 2 and Appendix B should also be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Xinyi Li, Jialu Shi, and Qiheng Xu have a real result here. For random walk on the d-dimensional torus, d≥3, they prove that P(C_N ≤ γ t_cov) = exp(-(1+o(1)) N^{d(1-γ)}) for γ > (d+2)/(2d), and a matching-order lower bound for all γ∈(0,1). The upper bound was known; the sharp constant -1 and the lower-bound machinery are new. The loop-insertion surgery in Section 4 is a genuinely novel proof ingredient, and the deterministic recovery of the original path from the modified one is clever. The paper is also honest: it states the range restriction, says what is sketched, and does not overclaim.\n\nThe main theorem is proved in detail along the coupling route from Prévost-Rodriguez-Sousi. The sharp range γ > (d+2)/(2d) is exactly where the coupling error exp(-cρ√(uN^{d-2})) is negligible against exp(-N^{d(1-γ)}), and the authors say so. That part is solid.\n\nThe soft spot is real, and it is the one the stress-test note identifies. Lemma 1.7, a generalization of [26, Lemma 6.1] to the N-dependent late-point regime α_N = γ - K/(d g(0) log N), is asserted with 'still goes through' and no proof. This lemma controls the one- and two-point probabilities of late points uniformly in α and in the set G. Lemma 1.9, Lemma 1.8, and Lemma 3.4 all rest on it, and through them the entire lower-bound event E. If the asserted uniformity fails at the stated rate, the o(1) in Lemma 1.9 could acquire a logarithmic factor and the concentration of late points would degrade. The stress-test note is not a hypothetical; it lands on the actual weakest link. The authors flag the gap themselves, but it is load-bearing, not cosmetic.\n\nThe other shortcuts, Theorem 0.2's sketch with Lemma 3.8 omitted and Appendix B's sketch of the interlacements counterpart, do not feed into the main sharp theorem, so I treat them as minor.\n\nBottom line: if Lemma 1.7 is true, this is a complete and important proof. The structure is coherent, and the honest presentation earns the benefit of the doubt. Send it to a serious referee; the referee should demand a full proof of Lemma 1.7 (or at least a rigorous uniformity statement) and a fuller write-up of Lemma 3.8. I would not accept the current version as final without that. But it absolutely deserves referee time, and I expect the result to stand.","headline":"Likely resolves the conjectured large-deviation exponent for torus cover times in d≥3, with a new sharp asymptotic, but one load-bearing lemma is asserted rather than proved.","tokens_in":38229,"tokens_out":2541,"would_cite":true,"duration_ms":21689,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C81","60F10","60G70"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for a simple random walk on $(\\mathbb{Z}/N\\mathbb{Z})^d$ with $d\\ge 3$, the probability of covering the torus by time $\\gamma$ times its expected cover time is $\\exp(-(1+o(1))N^{d(1-\\gamma)})$ for…","keywords":["random walk","cover time","large deviations","random interlacements","discrete torus","late points","loop insertion","strong coupling"],"falsifier":"For $d=3$ and $\\gamma=0.9$ (above the sharp-range threshold $5/6$), estimate $\\log P(C_N\\le 0.9\\,t_{\\rm cov})/N^{0.3}$ by Monte Carlo for $N=20,40,60$; the theorem predicts this ratio approaches $-1$, so a systematic deviation would contradict the claimed asymptotic. On the theoretical side, proving or refuting the stronger coupling error $\\exp(-c\\rho^2 uN^{d-2})$ would settle whether the sharp range can be extended to $\\gamma>2/d$.","tokens_in":37001,"feed_emoji":"🎲","tokens_out":10121,"duration_ms":79660,"temperature":0.7,"pith_summary":"Simple random walk on the discrete torus $(\\mathbb{Z}/N\\mathbb{Z})^d$, $d\\ge 3$, typically covers every site by time $t_{\\rm cov}=g(0)N^d\\log(N^d)$. This paper studies how unlikely it is to finish noticeably earlier, at time $\\gamma t_{\\rm cov}$ for $\\gamma\\in(0,1)$. Its main theorem shows that for $\\gamma\\in((d+2)/(2d),1)$ the probability is $\\exp(-(1+o(1))N^{d(1-\\gamma)})$; combined with the existing upper bound, the same exponential order $\\exp(-N^{d(1-\\gamma)+o(1)})$ holds for every $\\gamma\\in(0,1)$. The proof supplies sharp large-deviation lower bounds by coupling the random walk trace to random interlacements and by inserting recoverable loops into the trajectory to cover late points near the edge.","feed_headline":"Cover-time shortfall on tori costs exp(-N^{d(1-γ)})","feed_subtitle":"Sharp rates for d≥3 match the predicted exponent across the full large-deviation range.","key_machinery":"The machinery is the strong coupling between the trace of a simple random walk on the torus up to time $uN^d$ and the trace of random interlacements at level $u$ in the same region (Proposition 1.5). Random interlacements are a Poissonian cloud of bi-infinite simple random walk trajectories on $\\mathbb{Z}^d$, with the property that the probability a finite set $K$ is avoided is $\\exp(-u\\,\\mathrm{cap}(K))$. The coupling gives simultaneous inclusions $I^{u(1-\\rho)}\\cap Q_\\delta \\subset X[0,uN^d]\\cap Q_\\delta \\subset I^{u(1+\\rho)}\\cap Q_\\delta$, up to an error term $C N^{2d}\\lceil uN^{d-2}\\rceil\\exp(-c\\rho\\sqrt{uN^{d-2}})$. The lower-bound proof then combines this coupling with Harris-FKG to cover the bulk late points and with a deterministic loop-insertion surgery to cover the edge late points, adding loops whose total length is of order $N^{d(1-\\gamma)}$ and which can be uniquely deleted from the modified trajectory.","core_discovery":"The central claim is Theorem 0.1: for $\\gamma\\in((d+2)/(2d),1)$, $$\\lim_{N\\to\\infty}\\frac{\\log P(C_N\\le \\gamma t_{\\rm cov})}{$N^{{d(1-\\gamma)}}$}=-1,$$ where $C_N$ is the cover time of the torus and $t_{\\rm cov}=g(0)N^d\\log N^d$. Equivalently, $P(C_N\\le \\gamma t_{\\rm cov})=\\exp(-(1+o(1))N^{d(1-\\gamma)})$. The paper also proves a matching lower bound for the whole range $\\gamma\\in(0,1)$ (Theorem 0.2), which together with the known upper bound yields the full-rate statement $P(C_N\\le \\gamma t_{\\rm cov})=\\exp(-N^{d(1-\\gamma)+o(1)})$ for all $\\gamma\\in(0,1)$ (Corollary 0.3).","pith_inferences":["If the strong-coupling error were improved to $\\exp(-c\\rho^2 uN^{d-2})$, the same strategy would extend the sharp range from $\\gamma>(d+2)/(2d)$ down to $\\gamma>2/d$; the paper identifies this error as the main obstacle.","The rate constant 1 in the exponent is consistent with a simple heuristic: at the cover threshold there are of order $N^{d(1-\\gamma)}$ untouched sites, each missed with probability about $e^{-1}$, giving the same exponential rate as independent miss events.","The loop-insertion surgery is a general combinatorial device that could also be applied to the maximal-local-time large-deviation question on $\\mathbb{Z}^d$ raised in the paper, where the analogous conjecture is $\\exp(-N^{1-\\gamma+o(1)})$."],"forward_implications":["For $\\gamma\\in((d+2)/(2d),1)$, the probability of early cover has the sharp asymptotics $\\exp(-(1+o(1))N^{d(1-\\gamma)})$.","For every $\\gamma\\in(0,1)$, $P(C_N\\le \\gamma t_{\\rm cov})=\\exp(-N^{d(1-\\gamma)+o(1)})$, closing the exponent for the full large-deviation regime.","Upward deviations satisfy $P(C_N\\ge \\gamma t_{\\rm cov})=(1+o(1))N^{-d(\\gamma-1)}$ for $\\gamma>1$.","For random interlacements on a cube, the paper sketches the analogous sharp rate $P(M_N\\le \\gamma u_N)=\\exp(-(1+o(1))N^{d(1-\\gamma)})$ for $\\gamma\\in(2/d,1)$."],"supporting_citations":[{"why":"Supplies the strong coupling of random walk trace and random interlacements with the explicit error term used throughout the proofs.","marker":"[26]"},{"why":"Gives the Brownian upper bound for early cover that leads to the conjecturally sharp exponent after discrete adaptation.","marker":"[18]"},{"why":"Provides the discrete upper bound and the earlier 2D large-deviation strategy whose exponents the present paper matches.","marker":"[12]"},{"why":"Establishes the Gumbel fluctuation of the cover level for random interlacements, used to compute the e^{-1} cover probability of mesoscopic cubes.","marker":"[5]"},{"why":"Gives the first-order cover-time asymptotics and the late-point structure for the torus that the argument builds on.","marker":"[6]"},{"why":"Introduces random interlacements, the model whose capacity formulas and vacant-set probabilities carry the key estimates.","marker":"[27]"},{"why":"Earlier macroscopic coupling of torus random walk and random interlacements, a direct precursor of the strong coupling result.","marker":"[32]"}],"fun_headline_variants":["Torus cover time: exact large-deviation exponent for d≥3","Quick cover on high-d torus: rate exp(-N^{d(1-γ)})","Cover-time lower tail on tori: exponent N^{d(1-γ)} for d≥3","Rare short covers of tori: sharp rate at γ>(d+2)/(2d)","d≥3 torus: fast cover probability decays as exp(-N^{d(1-γ)})"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on an approximation of the random walk's path by a Poissonian cloud of infinite random walk trajectories, and the approximation must be accurate enough that its error is negligible compared with the tiny probability being computed; this accuracy only holds when $\\gamma$ is larger than $(d+2)/(2d)$.","fun_headline_variants_meta":{"raw":{"variants":["Torus cover time: exact large-deviation exponent for d≥3","Quick cover on high-d torus: rate exp(-N^{d(1-γ)})","Cover-time lower tail on tori: exponent N^{d(1-γ)} for d≥3","Rare short covers of tori: sharp rate at γ>(d+2)/(2d)","d≥3 torus: fast cover probability decays as exp(-N^{d(1-γ)})"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001582,"raw_usage":{"total_tokens":6297,"prompt_tokens":918,"completion_tokens":5379,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":5257}},"tokens_in":534,"tokens_out":5379,"duration_ms":38701,"temperature":1.0,"reasoning_tokens":5257,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:09:41.383364+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $d=3$ and $\\gamma=0.9$ (above the sharp-range threshold $5/6$), estimate $\\log P(C_N\\le 0.9\\,t_{\\rm cov})/N^{0.3}$ by Monte Carlo for $N=20,40,60$; the theorem predicts this ratio approaches $-1$, so a systematic deviation would contradict the claimed asymptotic. On the theoretical side, proving or refuting the stronger coupling error $\\exp(-c\\rho^2 uN^{d-2})$ would settle whether the sharp range can be extended to $\\gamma>2/d$.","supporting_citations":[{"cited_title":"Phase transition for the late points of random walk","cited_arxiv_id":"2309.03192","evidence_quote":"Supplies the strong coupling of random walk trace and random interlacements with the explicit error term used throughout the proofs."},{"cited_title":"and DEN HOLLANDER , F","cited_arxiv_id":null,"evidence_quote":"Gives the Brownian upper bound for early cover that leads to the conjecturally sharp exponent after discrete adaptation."},{"cited_title":"and V ACHKOVSKAIA , M","cited_arxiv_id":null,"evidence_quote":"Provides the discrete upper bound and the earlier 2D large-deviation strategy whose exponents the present paper matches."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Gumbel fluctuation of the cover level for random interlacements, used to compute the e^{-1} cover probability of mesoscopic cubes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the first-order cover-time asymptotics and the late-point structure for the torus that the argument builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces random interlacements, the model whose capacity formulas and vacant-set probabilities carry the key estimates."},{"cited_title":"and T EIXEIRA , A","cited_arxiv_id":null,"evidence_quote":"Earlier macroscopic coupling of torus random walk and random interlacements, a direct precursor of the strong coupling result."}],"review_version":1}