{"id":"af2de018-c24f-484d-8a97-ecaba5f4ae9c","arxiv_id":"2506.16873","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For i.i.d. perturbations of Z^d with mild tail conditions, there exists a translation-invariant perfect matching whose distance tail is bounded by a power of the hole probability.","lead":"Randomly perturbing every point of a lattice produces a random point cloud. This paper constructs a translation-invariant way to pair every cloud point back to a lattice point, with the pairing distance tail matching the natural lower bound from empty holes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The compactness/averaging step in §2.4 is under-specified: it does not prove that the weak limit of averaged matchings remains an injective matching into Π, and the subsequent perfectness argument assumes exactly that.","rationale":"The reader's weakest_assumption focused on (Int), but (Int) is an explicit assumption of Theorem 1.1 and its failure for α ≤ 1 is acknowledged and treated separately in Theorem 1.2 and Section 4.3, so it is not a hidden gap in the central claim. The reader's rationale does, however, flag the preservation of injectivity under the weak limit, which is the same region of the proof I consider most load-bearing. I do not believe the concern is fatal: for a.s. locally finite simple point processes, the set of pairs (Π, M) with M an injective matching into Π is closed under vague convergence of Π and coordinatewise convergence of M, because isolated points of a locally finite set cannot be the limit of multiple distinct points. With that lemma added, the averaging argument yields an invariant matching into the correct point process, the tail bound is preserved by upper semicontinuity of the closed tail event, and the mass transport argument proves perfectness. Thus the gap is technical rather than conceptual, and it does not invalidate the central claim. Other flagged issues — the d=1 atomic greedy matching and minor summations — are likewise repairable and outside Theorem 1.1. My verdict therefore remains CONDITIONAL, the same as the reader's, so no change is needed.","tokens_in":13273,"tokens_out":41231,"duration_ms":429691,"concrete_test":"Write down and prove the following closedness lemma, or find a counterexample: Let (Π_n, M_n) be a sequence of pairs in which Π_n is a locally finite simple point configuration, M_n : Z^d → Π_n is injective, and (Π_n, M_n) converges in the vague topology on configurations and coordinatewise on assignments. Then the limiting M satisfies M(u) ∈ Π for every u, and M is injective almost surely. If the lemma is true, the averaging step in §2.4 is valid; if a counterexample exists, the construction of a perfect invariant matching in Theorem 1.1 is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.1 requires a perfect invariant matching M′ : Z^d → Π with tail bound P(∥M′(0)∥ ≥ r) ≤ h(r)^c. The proof first produces, for each realization, a matching M from Proposition 2.1 with M(v) ∈ N_v, but M is not invariant and is not asserted to be perfect. Invariance is obtained in §2.4 by averaging the law μ of (Π, M) over lattice translations, μ_n := n^{-d} Σ_{v∈[n]^d} T_v μ, and taking a weak limit ν. The proof then invokes the mass transport principle to show that the limiting matching is perfect. This step is load-bearing and under-justified in two ways. First, T_v M is a matching into the translated point set Π + v, not into the original Π; the weak limit of these averaged laws is taken on the space of pairs (Π, M), and it is not automatic that the limiting M is a matching into the limiting Π in the sense needed for Υ(u,v) = P(M′(u) = Π_v). Second, even if M_n(u) ∈ Π_n for each n and M_n is injective, injectivity and the membership relation are not obviously closed in the relevant topology when Π_n varies; a limit could conceivably send two lattice points to the same limit point or to a point outside the limit configuration. The subsequent mass transport argument presupposes that M′ is already a matching into Π, so if the limiting object is only a transport plan or a matching into a different copy of the process, Theorem 1.1 does not follow. The paper cites a 'standard averaging argument' but supplies no closedness or measurability lemma for the space of matchings. This is the most load-bearing gap because every other step — the deterministic cover, the tail estimates for R_v, and the hole-probability comparison — appears sound under (Int) and (Reg).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for a randomly perturbed lattice Π={v+ξ_v: v∈Z^d} with i.i.d. perturbations satisfying regularity assumptions (Int) and (Reg), a translation-invariant perfect matching from Z^d to Π whose matching-distance tail is bounded by a power of the hole probability h(r)=P(Π∩B_r=∅). The proof combines a deterministic Hall-type matching criterion based on a random dyadic cover, probabilistic estimates for the tail of the cover radii, and a compactness/averaging argument to obtain invariance. In dimension one, for polynomial perturbations with tail exponent α∈(0,1), the paper proves the existence of a perfect matching with tail O(r^{-(1+α)/2}) and a matching lower bound showing that the moment threshold E[|M(0)|^{(1+α)/2}]=∞ is sharp.","tokens_in":13684,"tokens_out":16348,"duration_ms":152093,"significance":"The main theorem is a substantial positive result: it shows that the hole probability, which is the natural lower bound for any matching, is attainable up to a power for a large class of perturbed lattices, including Gaussian perturbations in all dimensions, where the tail becomes exp(-c r^{d+2}). The deterministic cover lemma and the hole-probability estimates are elegant, and the one-dimensional Theorem 1.2 is sharp and identifies a genuine transition for infinite-mean perturbations. The authors are appropriately explicit that the constructed matching is not a factor and that the factor version remains open. The paper is well organized and the overall strategy is convincing; however, the compactness step that produces the invariant perfect matching and several inequalities in Lemma 2.6 need to be repaired before the proof is complete.","major_comments":[{"comment":"The compactness/averaging step is load-bearing, because it is what turns the non-invariant matching of Corollary 2.4 into an invariant perfect matching M′: Z^d → Π, and as written it is not justified. The measures μ_n are averaged under translations that act on both the matching and the configuration, so T_v μ is a law of a matching from Z^d into Π+v, not into Π. The weak limit ν is asserted to be a law of a matching M′: Z^d → Π, but no topology is specified in which the requirements “M′(u) is one of the points of Π” and “u ↦ M′(u) is injective” are closed under weak convergence. Without such a closedness lemma, the subsequent mass-transport argument, which presupposes that M′ is a genuine matching into Π, does not apply. Please supply a precise compact space of matchings, or an alternative argument such as an exhaustion by finite-volume invariant matchings with a coupling that preserves membership, and prove that the limiting object is a perfect matching into Π.","section":"§2.4 (Proof of Theorem 1.1)"},{"comment":"The display following (2.10) asserts that Σ_{A⊆L0} P(∀v∈A, v+ξ_v∉Q_r) ≤ 2^{r^d} p(εr)^{r^d/4} (Reg) ≤ p(r)^{c r^d}. The appeal to (Reg) is problematic because (Reg) is stated for arguments k r with k≥1, whereas here ε<1 and p(εr) ≥ p(r). For Gaussian tails p(εr)=p(r)^{ε^2}, and for polynomial tails p(εr) is only a constant multiple of p(r); in both cases the desired bound holds only if c is chosen sufficiently small (e.g. c<ε^2/4 for Gaussians and c<1/4 for polynomials), which is not stated. Please state the choice of c and its dependence on ε explicitly, or replace (Reg) by a two-sided regularity condition valid for both r and εr.","section":"§2.3, Lemma 2.6, inequality (2.10)"},{"comment":"Two further inequalities in Lemma 2.6 are under-justified as written. In the bound on P(R1_v>r), the passage from the union bound to p(r)^{c r^d} requires proving that Σ_{n≥0} C n^{d-1} p(r)^{c[(r+n/4)^d-r^d]} is bounded for large r; this is true under the stated assumptions but is not shown. In the following display, the event {R_v>r} yields the existence of u with R1_u>r/2 and ∥v-u∥≤R1_u, so the relevant probability is P(R1_u>max(r/2,n)), not P(R1_u>min(r/2,n)); with “min” the small-n terms cannot be controlled by p(r)^{c r^d}. Please correct the extremum and justify the convergence of the sum.","section":"§2.3, Lemma 2.6 (tail of R1_v and R_v)"}],"minor_comments":[{"comment":"In the estimate for N0, the event should be v+ξ_v∉Q_r (the box being crossed), not v+ξ_v∉B_r; B_r was defined as the ℓ∞ ball and is not relevant for an arbitrary box Q_r.","section":"§2.3, Lemma 2.6, display after (2.9)"},{"comment":"The sentence “Let μ be the probability measure on (R^d)^{Z^d} corresponding to the above matching M” does not specify how a matching is encoded in that state space; please define the state space as pairs (Π, f) with f: Z^d → Z^d describing M(v)=Π_{f(v)}.","section":"§2.4"},{"comment":"The statement says “There is a perfect matching M” but the proof constructs an invariant stable matching; please state translation invariance explicitly in the theorem.","section":"Theorem 1.2 (1)"},{"comment":"The proof of Hall's condition counts N(A) as a multiset, but the final perfectness argument says “Since Π is countable”; for perturbation laws with atoms, Π is a multiset, so countability should be interpreted as countable support with multiplicities.","section":"§2.1, Proposition 2.1 and §2.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is worth publishing in math.PR if the authors provide a rigorous treatment of the averaging/compactness step in §2.4 and fix the inequalities in Lemma 2.6. The issues appear fixable within the scope of the manuscript, so I do not recommend rejection; I would be willing to review a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know before reading: the main theorem is almost certainly right, but the proof has a gap in exactly the place the stress-test points to. The paper constructs, for i.i.d. perturbations of Z^d satisfying (Int) and (Reg), a non-invariant matching M with tail P(||M(0)|| >= r) <= h(r)^c. That part is solid: the deterministic cover-crossing criterion (Prop 2.1), the box construction (Lemmas 2.3-2.4), and the estimates on R_v (Lemma 2.6) all line up. The hole probability asymptotics in Lemma 2.5 are clean. This is genuinely new: prior work covered Poisson and GEF zeros, not i.i.d. perturbed lattices, and the cover method is a nice contribution. The d=1 heavy-tailed result (Theorem 1.2) is also interesting and the lower bound seems right.\n\nThe soft spot is §2.4. The averaging argument produces an invariant law ν as a weak limit of averaged matchings. The stress-test's first worry—that the limit M' may not map into the limiting Π—is actually not a problem: membership is closed in the relevant topology, so if M_n(v) ∈ Π_n and the pairs converge, then M'(v) ∈ Π. The real issue is injectivity. The function space with pointwise convergence does not have closed injectivity; a limit can send two lattice points to the same point even if every approximating matching is injective. The mass transport principle then only gives expected preimage count 1, which doesn't rule out collisions. The paper asserts 'this gives an invariant matching' without proving that the limit is injective. That is a genuine gap in the proof of Theorem 1.1. I agree with the reader that it is repairable—probably by working with the matching as a random edge set and a finer topology, or by using a stable matching construction—but as written it is load-bearing.\n\nOther smaller issues: the d=1 greedy matching in the atomic case is hand-waved; the existence of a stable matching with ties is not fully proved. A few sums in Lemma 2.6 are sketchy but look fixable. None of these threaten the overall picture.\n\nWho is this for? Anyone working on invariant matchings, hyperuniform point processes, or optimal transport tail behavior. It deserves a serious referee: the result is new, the method is original, and the gap is specific and likely fixable. I would send it out, with a request to the authors to address §2.4 carefully.","headline":"Solid new result with a real gap in the compactness step; the main theorem is likely true and the paper deserves refereeing, but §2.4 needs work.","tokens_in":14223,"tokens_out":7487,"would_cite":true,"duration_ms":82469,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60D05","60G55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a randomly perturbed lattice can be matched back to the integer lattice by a translation-invariant perfect bijection whose matching-distance tail is bounded by a power of the hole probability, so the natural lower…","keywords":["random perturbed lattice","translation-invariant perfect matching","matching distance tail","hole probability","hyperuniform point process","Gaussian perturbation","polynomial perturbation","stable matching"],"falsifier":"Take $d=1$ with symmetric polynomial tails $P(|\\xi|\\ge r) \\asymp r^{-1/2}$: the hole probability is $\\exp(-\\Theta(r \\log r))$, while part (2) of Theorem 1.2 proves that every invariant perfect matching has $E|M(0)|^{3/4}=\\infty$; checking this divergence directly through the tail of $M(0)$ would confirm that the main theorem's optimal-tail conclusion cannot hold for heavy-tailed perturbations, delimiting exactly where the claim stops.","tokens_in":13074,"feed_emoji":"🎲","tokens_out":12789,"duration_ms":124386,"temperature":0.7,"pith_summary":"The paper proves that a randomly perturbed lattice—the integer lattice with each site shifted by an independent random vector—can be matched back to the lattice by a translation-invariant perfect bijection whose matching distance has the same tail decay as the probability that a large ball contains no perturbed point, up to a constant in the exponent. That hole probability is the natural lower bound: if a ball is empty, the matched image of the origin must lie outside it. The main theorem establishes this for all perturbation laws satisfying two regularity assumptions, covering Gaussian perturbations in every dimension and polynomial perturbations with decay faster than $r^{-1}$. For one-dimensional heavy-tailed polynomial perturbations, the paper shows optimality has a different form: an upper tail of order $r^{-(1+\\alpha)/2}$ and a matching lower bound on all invariant matchings.","feed_headline":"Jittered lattices admit perfect matchings with optimal tails","feed_subtitle":"Matching tails now provably decay as fast as the hole-probability lower bound allows.","key_machinery":"The engine is a deterministic matching criterion plus a random multi-scale cover. A regular cover $\\mathcal{D}$ partitions $\\mathbb{R}^d$ into boxes aligned with the lattice; a lattice site $v$ crosses a box if the segment from $v$ to its own perturbed point $\\Pi_v$ meets the box without ending inside it. Proposition 2.1 shows that if every box $D$ satisfies $|C(D)| \\le |D \\cap \\mathbb{Z}^d|$, then the standard bipartite matching criterion yields a matching with $M(v)$ lying in $D_v$ or an adjacent box, so the matching distance is bounded by the local box scale. The paper then builds a dyadic box cover whose scale $R_v$ is the first level at which the crossing count fits inside the box's lattice points, smoothed so neighbouring scales vary slowly, and proves probabilistically that $P(R_v > r) \\le h(r)^c$ using tail estimates for crossing points via the integrability and regularity assumptions. A compactness-and-averaging argument converts the resulting random matching into a translation-invariant one, and the mass-transport principle shows it is perfect. The one-dimensional heavy-tailed construction instead uses a greedy stable matching together with variance estimates for the counting function.","core_discovery":"Let $\\Pi = \\{v + \\xi_v : v \\in \\mathbb{Z}^d\\}$ with i.i.d. random vectors $\\xi_v$, and let $p(r)=P(\\|\\xi\\|\\ge r)$ and $h(r)=P(\\Pi \\cap B_r = \\emptyset)$. The paper's main claim is that, whenever the tail $p$ satisfies the integrability condition $\\int_r^\\infty p(t)\\,dt \\le C r p(r)$ and the regularity condition $\\sup_{r\\ge 1} \\log p(kr)/\\log p(r) < \\infty$, there exists a translation-invariant perfect matching $M : \\mathbb{Z}^d \\to \\Pi$ with $P(\\|M(0)\\| \\ge r) \\le h(r)^c$ for all large $r$. In words, the matching distance tail is no worse than a power of the hole-probability tail, so the natural lower bound is attained up to the constant $c$ in the exponent. For Gaussian perturbations this gives $P(\\|M(0)\\| \\ge r) \\le \\exp(-c r^{d+2})$; for polynomial perturbations with $\\alpha>1$ it gives $\\exp(-c r^d \\log r)$. In $d=1$ with polynomial tails of exponent $\\alpha\\in(0,1)$, the paper instead constructs a factor matching with tail $r^{-(1+\\alpha)/2}$ and proves that every invariant perfect matching has infinite moment of order $(1+\\alpha)/2$, so no faster algebraic tail is possible.","pith_inferences":["[Editorial extension] The cover-and-crossing criterion is a general device: perturbing any lattice-like point process by i.i.d. noise and checking the crossing-count condition at every dyadic scale gives a route to matching tails at the hole-probability scale for other hyperuniform ensembles, such as the eigenvalue process of a random Gaussian matrix, where the paper leaves existence open.","[Editorial extension] The non-factor character of the main construction suggests that private randomness may be essential for the optimal tail in $d\\ge 2$; a natural test is whether a deterministic factor matching can achieve the same bound for Gaussian perturbations, which the paper poses as an open problem.","[Editorial extension] The $d=1$ heavy-tailed results indicate a heuristic that in one dimension the matching tail is governed by fluctuations of the counting function, whereas in higher dimensions collective hole events dominate; if true, the conjectured absence of a transition in $d\\ge 2$ would follow from the fact that hole probability is far smaller than single-point displacement for all polyno","[Editorial extension] The regularity condition is used only to simplify the hole-probability comparison; the proof's remark suggests a weakened replacement. A testable project is to identify the maximal class of perturbations for which the crossing-scale tail is comparable to $h(r)$ without the regularity condition."],"forward_implications":["For any Gaussian perturbation of $\\mathbb{Z}^d$, an invariant perfect matching exists whose matching distance tail decays like $\\exp(-c r^{d+2})$, matching the hole-probability lower bound up to a constant in the exponent.","For polynomial perturbations with tail exponent $\\alpha>1$, the same construction gives tail $\\exp(-c r^d \\log r)$, again optimal up to the exponent constant.","The construction answers a question from the hyperuniform matching literature: two-dimensional Gaussian perturbed lattices admit invariant matchings with much faster than finite-moment tail decay.","In one dimension with heavy-tailed polynomial perturbations $\\alpha\\in(0,1)$, a factor matching achieves tail $r^{-(1+\\alpha)/2}$, and no invariant perfect matching can have finite $(1+\\alpha)/2$ moment.","The main matching is invariant and perfect but not a factor; whether a factor matching with the same tail exists is left open."],"supporting_citations":[{"why":"supplies the mass-transport principle used to prove the averaged matching is perfect, and the stable-matching framework used in the one-dimensional construction.","marker":"[12]"},{"why":"frames the hyperuniform matching problem and provides the prior finite-second-moment result that the main theorem strengthens.","marker":"[14]"},{"why":"supplies the concentration inequality used to bound counting-function fluctuations in the one-dimensional heavy-tailed construction.","marker":"[4]"},{"why":"supplies the central-limit theorem used to prove the lower-bound moment for all invariant matchings in one dimension.","marker":"[6]"},{"why":"supplies the stable-matching algorithm that defines the greedy matching in the one-dimensional construction.","marker":"[10]"}],"fun_headline_variants":["Matching tails hit hole-probability bound","Perfect matchings reach optimal decay","Optimal matching tails via hole probability","Invariant matchings: tails meet lower bound","Perturbed lattices: optimal matching tails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the perturbation tail decays fast enough that the integral of the tail from $r$ onward is at most a constant multiple of $r$ times the tail at $r$; when this fails, as for polynomial tails with $\\alpha\\le 1$, the theorem's conclusion is no longer claimed and the one-dimensional results show a genuinely different behavior.","fun_headline_variants_meta":{"raw":{"variants":["Matching tails hit hole-probability bound","Perfect matchings reach optimal decay","Optimal matching tails via hole probability","Invariant matchings: tails meet lower bound","Perturbed lattices: optimal matching tails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00021,"raw_usage":{"total_tokens":1394,"prompt_tokens":911,"completion_tokens":483,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":417}},"tokens_in":527,"tokens_out":483,"duration_ms":4818,"temperature":1.0,"reasoning_tokens":417,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:19:05.148707+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $d=1$ with symmetric polynomial tails $P(|\\xi|\\ge r) \\asymp r^{-1/2}$: the hole probability is $\\exp(-\\Theta(r \\log r))$, while part (2) of Theorem 1.2 proves that every invariant perfect matching has $E|M(0)|^{3/4}=\\infty$; checking this divergence directly through the tail of $M(0)$ would confirm that the main theorem's optimal-tail conclusion cannot hold for heavy-tailed perturbations, delimiting exactly where the claim stops.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the mass-transport principle used to prove the averaged matching is perfect, and the stable-matching framework used in the one-dimensional construction."},{"cited_title":"Chung and L","cited_arxiv_id":null,"evidence_quote":"supplies the concentration inequality used to bound counting-function fluctuations in the one-dimensional heavy-tailed construction."},{"cited_title":"Gale and L","cited_arxiv_id":null,"evidence_quote":"supplies the stable-matching algorithm that defines the greedy matching in the one-dimensional construction."}],"review_version":2}