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Optimal matchings of randomly perturbed lattices

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.16873 v1 pith:QO47UXJ3 submitted 2025-06-20 math.PR

Dor Elboim , Yinon Spinka , Oren Yakir This is my paper · ORCID
classification math.PR MSC 60D0560G55
keywords randomperturbedlatticetranslation-invariantperfectmatchingdistancetailholeprobabilityhyperuniformpointprocessGaussianperturbationpolynomialstable
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • [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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§2.4 (Proof of Theorem 1.1)] 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.
  2. [§2.3, Lemma 2.6, inequality (2.10)] 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.
  3. [§2.3, Lemma 2.6 (tail of R1_v and R_v)] 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.
minor comments (4)
  1. [§2.3, Lemma 2.6, display after (2.9)] 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.
  2. [§2.4] 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)}.
  3. [Theorem 1.2 (1)] 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.
  4. [§2.1, Proposition 2.1 and §2.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the tail bound is derived from independently defined hole probabilities and matching estimates, not from fitted inputs or self-citations.

full rationale

No circular step is present. The main theorem (Theorem 1.1) asserts a tail bound on the matching distance in terms of h(r) = P(Π ∩ B_r = ∅), which is an independently defined quantity. The proof proceeds by constructing a deterministic matching from a regular cover (Proposition 2.1 and Corollary 2.4), then proving probabilistic tail bounds on the cover radius R_v in Lemma 2.6. Lemma 2.5 independently establishes two-sided comparisons between h(r) and p(r) under (Int) and (Reg), and Lemma 2.6 uses those comparisons to convert its own p(r)-based estimate into the claimed h(r)^c bound. There is no fitted parameter that is later renamed as a prediction, and no definition of h(r) in terms of the matching tail. The matching tail is not used to define h(r). The self-citations ([1], [26], [27]) appear only in contextual remarks about related literature and do not carry any load-bearing argument; the mass transport principle is cited to [12], an external source. The averaging argument in Section 2.4 is compressed and may conceal a technical closure question about weak limits of matchings into translated copies of Π, but that is a potential proof gap rather than circularity: the conclusion is not assumed by construction. The paper also explicitly flags limitations and open problems (factor matching is left open, (Reg) is said to be non-essential, and the subcritical polynomial case in d ≥ 2 is conjectural), which further indicates that the derivation is not circular. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof relies on two explicit tail regularity assumptions, (Int) and (Reg), stated in Section 1.2. These are domain assumptions on the perturbation law, not fitted parameters. The paper also makes two under-proved assertions: the existence of the greedy stable matching in the d=1 atomic case and the preservation of injectivity in the averaging weak limit. No free parameters or invented entities are used.

assumptions (7)
  • domain assumption Perturbation tail p(r) satisfies (Int): ∫_r^∞ p(t) dt ≤ C r p(r) for all r ≥ 1.
    Stated in Section 1.2. Used in Lemma 2.6 to bound E[∥ξ∥ 1_{∥ξ∥≥εr}] ≤ C r p(r)^c and in Lemma 2.5 for the hole-probability lower bound.
  • domain assumption Perturbation tail p(r) satisfies (Reg): sup_{r≥1} log p(kr)/log p(r) < ∞ for all k ≥ 1.
    Stated in Section 1.2. Used in Lemmas 2.5 and 2.6 to compare p(r/2), p(εr), and p(3r) with p(r).
  • ad hoc to paper A greedy stable matching exists for Z and its perturbed copy Π when ξ has atoms, with left-most tie-breaking.
    Asserted in Section 3 without proof; the standard reference [12, Lemma 15] requires non-equidistance and no descending chains, which fail here.
  • ad hoc to paper The weak limit of averaged matching measures is supported on injective matchings.
    Invoked in Section 2.4; injectivity is not closed in the product topology, so the limit could in principle have collisions.
  • standard math Hall's marriage theorem for finite bipartite graphs with multisets.
    Used in Proposition 2.1.
  • standard math Mass transport principle for translation-invariant matchings.
    Cited from [12, Lemma 8] and used in Section 2.4 to prove the matching is perfect.
  • standard math Lindeberg-Feller central limit theorem and Freedman's inequality.
    Used in Section 3 for the variance bound and chaining argument.

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Pith. "Pith review of Optimal matchings of randomly perturbed lattices." pith.science (2026). https://pith.science/paper/QO47UXJ3

@misc{pith2026250616873,
  author       = {Pith},
  title        = {Pith review of: Optimal matchings of randomly perturbed lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QO47UXJ3}},
  note         = {Machine review of arXiv:2506.16873}
}
read the original abstract

Consider a point process in Euclidean space obtained by perturbing the integer lattice with independent and identically distributed random vectors. Under mild assumptions on the law of the perturbations, we construct a translation-invariant perfect matching between this point process and the lattice, such that the matching distance has the same tail behavior as the hole probability of the point process, which is a natural lower bound.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Sampling properties of the zeroes of the Gaussian entire function

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