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Rainbow percolation

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Pith's one-line read A dependent long-range percolation model on the line has a genuine phase transition, with β_c between 1 and 31.

desk verdict A genuine resolution of the excluded p=1 endpoint in one-dimensional long-range percolation, with a credible proof that the continuum model has beta_c in [1,31] while its lattice skeleton never percolates; deserves serious refereeing. read the letter →

arxiv 2608.12954 v1 pith:KHE4GLET submitted 2026-08-13 math.PR

Peter Gracar , Benjamin Lees This is my paper · ORCID
classification math.PR MSC 60K3560D0582B43
keywords long-rangepercolationcriticaldecayweight-dependentrandomconnectionmodelParetoradiiphasetransitionPoissonpointprocessthresholdrainbowconstruction
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 studies a one-dimensional random graph on a Poisson point process of intensity $\lambda$, where a point with mark $t$ behaves as a ball of radius $\beta/t$; two points are joined when each lies inside the other's ball. The connection probability between points at distance $d$ is $\min(1,\beta/d)^2$, the critical decay of one-dimensional long-range percolation, but edges are dependent because they share radii. The paper proves that this model has a genuine phase transition: for $\lambda\beta<1$ all components are finite almost surely, while for $\lambda\beta\ge 31$ an infinite component exists almost surely, so at unit intensity the critical value satisfies $\beta_c\in[1,31]$; a numerical study places it near 2. In contrast, the discrete version on $\mathbb{Z}$, obtained by pinning vertices to integer sites, is totally fragmented for every admissible parameter and has only a degenerate transition—evidence that the supercritical phase is a purely continuum effect driven by point density. This matters because the model sits at a boundary case of weight-dependent random connection models where prior criteria are silent, and the proof introduces two portable mechanisms: balanced 'rainbows' to certify disconnection and a dyadic hyperbolic renormalisation to certify percolation.

What carries the argument

The rainbow is a pair of nested edges $\{a,b\}$ and $\{c,d\}$ with $a<c<d<b$ whose diagonals $\{a,c\}$ and $\{d,b\}$ are missing and whose overhangs $\ell=c-a$, $m=d-c$, $r=b-d$ satisfy the balance condition $|\ell-r|\le m$. Balance ensures that the outer arch is the only edge allowed to cross a cut, so on a containment event the component of every vertex between two cut positions is trapped (Lemma 3.4); cut-point certificates are cap events whose probabilities are exact products, and Kochen–Stone plus the zero–one law turn uniform per-scale probability into almost sure confinement at infinitely many scales. For the supercritical direction, the mark space is split into dyadic bands $X_k$ with radii in $(\beta 2^k,\beta 2^{k+1}]$, each band being a dilated, dense bounded-range graph; the min-rule makes a chain of band-$k$ points absorb every larger-radius point in its span, and independent site percolation on the floored binary tiling of the hyperbolic half-plane, controlled by a Peierls contour bound using a boundary-connectivity lemma, yields the infinite cluster.

What would settle it

Simulate the model at $\lambda=1$, $\beta=0.9$ on a window of length $10^9$: Theorem 1 predicts the largest component fraction decays to zero with window size, so a reproducible spanning component would falsify it; symmetrically, at $\beta=31$ the construction predicts a percolating cluster, so observing none on large windows would falsify Theorem 2. A sharper check is to measure the per-scale probability of the balanced-rainbow event of Definition 3.1: it should stay uniformly positive in the scale, and if it decays to zero the lower-bound mechanism is broken.

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Extended reading notes

Core claim

The central claim is that the weight-dependent random connection model on $\mathbb{R}\times(0,1)$ with edge condition $(t\vee s)|x-y|\le\beta$ undergoes a non-degenerate phase transition. Theorem 1 states that when $\lambda\beta<1$, almost surely every connected component is finite; Theorem 2 states that when $\lambda\beta\ge 31$, an infinite connected component exists almost surely; together with the scaling to intensity one these give $1\le\beta_c\le 31$. Theorem 3 states that the discrete skeleton on $\mathbb{Z}$—vertices at integer sites carrying independent Pareto radii of scale $x_m$, joined when $|i-j|\le R_i\wedge R_j$—has no supercritical phase: for every $x_m\in(0,1)$ all components are finite, even though almost surely infinitely many edges cross every site, and the only transition is the degenerate one to full connectivity at $x_m\ge 1$. The paper also proves that infinitely many edges cross every point of the line almost surely, so fragmentation is achieved not by avoiding long edges but by confining every component between the cuts of rainbows.

Load-bearing premise

The subcritical bound rests on the assertion that balanced rainbows—nested edge pairs whose left and right overhangs differ by at most the inner span—form at every geometric scale with probability bounded below uniformly; if that block construction ever fails at a scale, the confinement argument and the lower bound $\beta_c\ge 1$ collapse.

Editorial extensions

If this is right

  • At unit intensity every $\beta<1$ gives only finite components, while every $\beta\ge 31$ gives an infinite component; the critical point is genuinely between 1 and 31, with numerics near 2.
  • The discrete skeleton shows that infinite components are not forced by the heavy-tailed radii alone: on $\mathbb{Z}$, with $x_m<1$, all components are finite and yet their diameters are unbounded, and infinitely many edges cross every site.
  • The boundary case $\gamma=0$ of the weight-dependent random connection model in one dimension is decided: a supercritical phase exists, closing a case where percolation-threshold criteria were silent.
  • The supercritical proof is quantitative: the site-open probability $(1-e^{-\lambda\beta/4})^{16}$ controls the failure rate, and optimising the contour constants would lower the 31 to about 15, still above the numerical critical value.
  • The numerical curves indicate a jump in percolation density at the threshold, with crossing heights 0.7–0.9, consistent with the known discontinuity for independent $1/|x-y|^2$ percolation.

Reading between the lines

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

  • The balance condition $|\ell-r|\le m$ appears to be the load-bearing geometric constraint; one could test whether relaxing it to $|\ell-r|\le cm$ for a large constant $c$ preserves a uniform per-scale confinement probability, which would pinpoint what the lattice skeleton's geometry contributes beyond the radius law.
  • The sharp contrast between the discrete and continuum models suggests that the same percolation mechanism should appear on any vertex set with unbounded local density per connection range, such as a Poisson process, but not on lattices; this is an editorial extrapolation, since the paper only proves the $\mathbb{Z}$ and $\mathbb{R}$ cases.
  • If the reported jump in the percolation density is confirmed rigorously, the model would extend the Aizenman–Newman discontinuity phenomenon from independent to dependent edges; proving or disproving that jump is a natural next step.
  • The dyadic renormalisation may carry to other one-dimensional models with scale-invariant connection kernels, since only the min-rule and the band density enter; testing it on a kernel with a different boundary exponent would show whether the mechanism is specific to Pareto radii.
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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper studies a weight-dependent random connection model on a Poisson point process of intensity λ on R×(0,1), with the deterministic edge rule (t∨s)|x−y|≤β, equivalently min-rule connection of Pareto radii with scale β. The main results are Theorem 1 (for λβ<1, almost surely every connected component is finite) and Theorem 2 (for λβ≥31, almost surely an infinite component exists), giving β_c∈[1,31] at intensity one. The lower bound is proved through a discrete skeleton on Z with i.i.d. Pareto radii, for which Theorem 3 establishes total fragmentation for every admissible scale x_m∈(0,1); the proof uses balanced rainbow events at independent scales, cut-point certificates with exact product probabilities, a confinement lemma, and a Kochen–Stone plus zero–one-law assembly. The continuum transfer replaces exact products by exact void probabilities of Poisson regions. The supercritical proof is a dyadic-band renormalization: independent band processes are glued deterministically by the min-rule, and the resulting site-percolation problem on a floored binary tiling is solved by a Peierls contour argument. A numerical appendix, clearly separated from the proofs, estimates β_c near 2.

Significance. If the results hold, this is a substantial contribution to dependent long-range percolation at the critical inverse-square decay. The paper gives a genuine phase transition in a model where edges through a common vertex are strongly dependent and where classical independent-edge results do not apply. The proofs are largely parameter-free and deliver explicit constants: the construction fixes 1 and 31 as rigorous bounds, with no fitted parameter entering the proofs. The discrete skeleton result is of independent interest and is surprising: total fragmentation coexists with almost surely infinitely many edges crossing every site. The supercritical argument uses only exact products over disjoint regions and deterministic gluing, avoiding correlation inequalities. The numerical study is clearly labeled as non-rigorous, and the ancillary code, seeds, and manifests make it reproducible; it is not used in any proof. The paper is careful to separate proven statements from numerical extrapolations, and its claims are falsifiable.

minor comments (5)
  1. [Section 2, proof of Proposition 2.1] In the displayed computation of E[X] after Eq. (1), the expression β^2/2+β^2∫_β^∞ dd/d appears to contain a typesetting artifact; it should read β^2/2+β^2∫_β^∞ d^{-1} dd.
  2. [Notation section] The glossary omits several later central objects, including the graph H and H⋆ of Section 5.3, the site events O_{k,i}, and the events D_k and E_k used throughout Sections 3 and 4; adding these would improve usability.
  3. [Figure 12, left panel] The caption states that the spanning threshold extrapolates to about 1.08, below β_c, without explaining that the per-sample spanning threshold is not a consistent estimator of β_c; the surrounding text explains this, but the caption alone is misleading.
  4. [Section 5.3, Lemma 5.6] The assertion that the bounded face boundaries generate the cycle space of H is stated without proof; since the application of Timár's lemma depends on it, a one-sentence justification for locally finite planar graphs, or a reference, would make the argument self-contained.
  5. [Remark 1.1(iv)] The phrase 'the dependence costs an exponent' could be misread: since x_m>x_m^2, the certificate exponent 1−x_m is smaller than the independent-pairs exponent 1−x_m^2, so the cost is a decrease of the exponent; the intended meaning is clear but the wording is terse.

Circularity Check

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No significant circularity: Theorems 1-3 are proved from explicit multi-scale constructions and arithmetic constants (1, 31); the numerical appendix is explicitly not used in the proofs.

full rationale

The derivation chain is self-contained. The lower bound rests on an explicit discrete-skeleton proof: rainbows are formed at independent scales with uniform probability (Proposition 3.1), cut-point certificates give gaps with probability bounded below (Propositions 3.2 and 4.1), and the Kochen-Stone plus zero-one-law assembly (Propositions 3.4 and 4.3) yields almost-sure confinement. No fitted parameter appears. The constant 1 enters only through the exponent 1-beta>0 in the cut-certificate count (Proposition 4.1), and 31 enters only through the elementary bound 16 e^{-31/4} <= 2^{-7} in Lemma 5.2; both are arithmetic, not tuned to data. The numerical appendix is expressly separated ('a numerical study included as an appendix places it near 2') and is not invoked in any theorem proof, so no fitted value is relabeled as a prediction. Self-citations ([14], [15], [17], [16]) appear solely as literature context or as remarks that prior criteria are silent on this boundary case (Remark 4.2), and are not load-bearing; the externally cited tools (Timár's boundary-connectivity lemma, Kochen-Stone, Last-Penrose) are standard and stated with their hypotheses. There is no uniqueness theorem imported from the authors and no ansatz adopted by self-citation. The discrete skeleton result (Theorem 3) is proved by its own nontrivial argument rather than assumed, and is cross-checked, not derived, against the classical independent-edge threshold in Remark 1.1. Honest non-finding: no circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard tools of probability theory and percolation theory, all cited and standard. No parameter is fitted to data; the proof constants (16, 31, etc.) are chosen arithmetically, not tuned to simulations. The numerical study is clearly separated from the proofs.

assumptions (6)
  • standard math Kolmogorov zero-one law
    Used in Propositions 3.4 and 4.3 to upgrade positive probability of {D_k i.o.} to probability one, after showing the event lies in the tail sigma-field.
  • standard math Kochen-Stone lemma
    Used in Propositions 3.4 and 4.3 to derive P(D_k i.o.)>0 from uniform lower bounds and controlled pairwise correlations.
  • standard math Harris-FKG inequality for Poisson processes (Last-Penrose Theorem 20.4)
    Lemma 4.1 uses it to lower-bound probabilities of intersections of decreasing events, a key ingredient in Propositions 3.3, 4.2 and Lemma 3.3.
  • standard math Timár's boundary-connectivity lemma
    Lemma 5.6(iii) applies it to the pair (H,H⋆) to obtain connectivity of the outer boundary of a finite open cluster, essential for the Peierls count.
  • standard math Ergodicity of the Poisson process under spatial translations
    Used at the end of Theorem 2 to argue the infinite-component event has probability 0 or 1.
  • standard math Poisson restriction theorem and marking theorem
    Used throughout Sections 4 and 5 to decompose the point process into independent bands and condition on positions.

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Pith. "Pith review of Rainbow percolation." pith.science (2026). https://pith.science/paper/KHE4GLET

@misc{pith2026260812954,
  author       = {Pith},
  title        = {Pith review of: Rainbow percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KHE4GLET}},
  note         = {Machine review of arXiv:2608.12954}
}
abstract

We consider the weight-dependent random connection model on a Poisson point process of intensity $\lambda$ on $\mathbb{R}\times(0,1)$ in which the vertices $(x,t)$ and $(y,s)$ are joined precisely when $(t\vee s)|x-y|\leq\beta$. Points at distance $d$ are joined with probability $\min(1,\beta/d)^2$, the critical decay of one-dimensional long-range percolation, and edges sharing a vertex are dependent through the common mark. We prove that the model has a genuine phase transition: for $\lambda\beta<1$ almost surely all connected components are finite, while for $\lambda\beta\geq 31$ an infinite component exists, so at intensity one the critical value satisfies $\beta_c\in[1,31]$; a numerical study included as an appendix places it near $2$. The lower bound is proved via a discrete skeleton of the model, obtained by pinning the vertices to $\mathbb{Z}$, which is of independent interest: it has no supercritical phase at all, jumping at a degenerate transition from total fragmentation to trivial connectivity, even though almost surely infinitely many edges cross every fixed site.

Figures

Figures reproduced from arXiv: 2608.12954 by the authors.

Figure 1
Figure 1. Lk = (−2 k , −2 k−1 ], Rk = [2k−1 , 2 k ) (to scale). Mk counts origin-crossing edges from Lk to Rk; distinct scales use disjoint vertices, so the Mk are independent, and Paley–Zygmund plus Borel–Cantelli give Mk ≥ 1 infinitely often. Proof. Let X be the number of origin-crossing edges, i.e. edges {(x, t),(y, s)} with x ≤ 0 < y. Averaging the connection indicator over the two independent uniform marks, two points at… view at source ↗
Figure 2
Figure 2. Two edges whose endpoints interleave, a < c < b < d. Because b − c is shorter than both b − a and d − c, the edge {b, c} is forced, so a ∼ b ∼ c ∼ d all lie in one component. Proof. Let {(a, ta),(b, tb)} and {(c, tc),(d, td)} be edges whose endpoints interleave, a < c < b < d, as in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. A rainbow: outer arch {a, b} over a nested inner arch {c, d}, with left overhang ℓ, inner span m, right overhang r. The non-edges {a, c}, {d, b} cap the inner radii (Rc < ℓ, Rd < r), forcing ℓ, r > m; balance |ℓ − r| ≤ m caps them further, by Rc < m + r and Rd < m + ℓ, so that the long diagonals {c, b} and {a, d} are absent too. 3 The simplified model To avoid lower order random effects, we work in this section with… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Four disjoint blocks at geometric scale g = 82k−1 , each of size g, host the candidate vertices a, c, d, b. Moderate inner radii ([4g, 5g]) force the inner arch {c, d} and forbid the short diagonals {a, c}, {d, b}; large outer radii (≥ 32g) force the outer arch {a, b}.…
Figure 5
Figure 5. Figure 5: If a vertex u has an edge leaving the window to some w /∈ [a, b], then Ru ≥ |u − w| > min(u − a, b − u), so u already attaches to an outer endpoint inside [a, b] (here {a, u}). Hence the connection question is settled by the radii in [a, b]. Proof. A path inside [a, b]…
Figure 6
Figure 6. Figure 6: On the containment event B the arches can meet only through an overhang, a ∼ c in (a, c) or d ∼ b in (d, b) (wavy). The dashed over-the-top route, an interior v with Rv ≥ v − a joining a to c across the span (and the long diagonal {a, d}), is what B forbids. On B and w…
Figure 7
Figure 7. Figure 7: Each interior vertex v = c+k sits at distance ℓ+k from a and r +m−k from b; it reaches an outer endpoint iff Rv ≥ Dk := min(ℓ + k, r + m − k). The event B that no interior vertex reaches out has P(B) = Q k (1 − xm/Dk) ≥ (4/9)2xm. Proof. Write Dk = min(ℓ + k, r + m − k)…
Figure 8
Figure 8. Figure 8: The inclusion C ♯ i ⊆ {gap♯ at i + 1}. A pair (j, k) with j ≤ i + 1 < k would have to span the dashed cut. The four dashed arcs exhaust the possibilities: j = 0, which splits according to whether k falls inside the overhang or beyond it, and the two remaining ranges of…
Figure 9
Figure 9. Figure 9: The joint bound. Cap profiles of C ♯ i (solid) and C ♯ i ′ (dashed) over the vertices 1, . . . , n − 1; each descends to its pinch at 1, then jumps back up and rises as v. The caps of the intersection are the vertex-wise minimum, so they follow the lower of the two cur…
Figure 10
Figure 10. Figure 10: Two scale-k windows inside their parent window; band-k points on the lower line (one per cell of length β2 k/2 on the site events), band-(k + 1) points on the upper line (one per parent cell of length β2 k ). Both cell grids are drawn: the parent cells are twice the c…
Figure 11
Figure 11. Figure 11: Vertices are the dyadic windows, drawn as the tiles [PITH_FULL_IMAGE:figures/full_fig_p035_11.png]
Figure 12
Figure 12. Figure 12: Median per-sample spanning threshold against 1 [PITH_FULL_IMAGE:figures/full_fig_p039_12.png]
Figure 13
Figure 13. Figure 13: The largest-component fraction θL(β) of the central subwindow, median over replicates, for window lengths L = 105 to 108 (darker with increasing L; 64 down to 4 replicates). The curves decay with L for β ≤ 1.5, are stable in L from β = 2 on, and sharpen with L while c…

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