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From Kreweras walks to branching perimeter processes of percolated triangulations

T0 review · 2 major / 6 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read The perimeter of percolation peeling on a triangulation equals the L1 distance of spine steps on its Kreweras walk to the walk’s endpoint.

desk verdict Clean discrete L1 identity linking Kreweras spine steps to peeling perimeters; short combinatorial note that genuinely bridges BHS and SPW25 without overclaiming. read the letter →

arxiv 2607.28566 v1 pith:NYUFR3PK submitted 2026-07-30 math.CO math-phmath.MPmath.PR

Renan Gross , Emmanuel Kammerer , Frederik Ravn Klausen This is my paper · ORCID
classification math.COmath-phmath.MPmath.PR MSC 05C1060C0560J8005A19
keywords Kreweraswalkspercolatedtriangulationspeelingprocessperimeterbranchingexplorationgrowth-fragmentationBernardibijectionmatingoftrees
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

This note proves an exact discrete identity: for a percolated planar triangulation with one white boundary vertex, the perimeter process recorded while peeling along the percolation interface is completely determined by the corresponding Kreweras walk. At each spine step of the walk (an a- or b-step not enclosed by a close matching), the perimeter equals the L1 distance from that point to the walk’s endpoint, plus two. The same geometric rule extends branch by branch to the full branching peeling exploration, using restricted spine times and the first later time the walk closes the cone containing that segment. The identity makes precise, at the discrete level, why growth-fragmentation perimeter processes appear inside correlated Brownian cone excursions in the continuum mating-of-trees picture.

What carries the argument

Spine steps of the Kreweras walk (and the associated reduced Kreweras word). Via the Bernardi–Holden–Sun bijection they are identified with the successive bicolour triangles traversed by the percolation interface; the height of each close-matched cone excursion becomes the size of the monochromatic face filled at an Lℓ or Rℓ peeling step, so perimeter increments match the L1 increments of the reduced walk.

What would settle it

Take any small explicit percolated triangulation with one white boundary vertex, compute its Kreweras walk by the known bijection, mark the spine steps, evaluate the claimed L1 formula, and check whether the resulting sequence equals the perimeter process obtained by peeling the map directly along its interface.

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Extended reading notes

Core claim

For a percolated triangulation with exactly one white boundary vertex and its Bernardi–Holden–Sun Kreweras walk X of length N, if T(n) marks the n-th spine step of X, then the interface perimeter process satisfies P(n) = ∥X(T(n+1)) − X(N)∥₁ + 2. The same L1 formula holds for every branch of the branching peeling exploration once spine times are taken inside the walk prefix that builds the target triangle and distances are measured to the cone-closure time of that prefix.

Load-bearing premise

The argument stands or falls on the already-proved correspondence that close matchings in the Kreweras word produce exactly the monochromatic faces of the spine; if that dictionary fails for some configurations, spine steps no longer line up with peeling steps.

Editorial extensions

If this is right

  • The discrete growth-fragmentation of branching perimeters is now an explicit functional of the Kreweras walk, not merely a scaling-limit object.
  • Joint scaling of the Kreweras walk to a Brownian cone excursion and of the branching perimeters to the self-similar growth-fragmentation becomes a question of local-time convergence of spine steps.
  • The same L1 reading extends immediately to Dobrushin boundaries with several consecutive black and white vertices.
  • An analogous dictionary can be asked for other walk encodings of statistical-mechanics decorations on maps (e.g., hamburger-cheeseburger).

Reading between the lines

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

  • Once spine-step local time is controlled, the three scaling limits (Kreweras walk, Brownian disk, growth-fragmentation) should converge jointly without further combinatorial input.
  • The first coordinate of X(T(n)) − X(N) literally counts white boundary vertices of the unexplored region, giving a direct colour-refined perimeter process.
  • The same geometric extraction may supply discrete precursors of the cone-free local times used in the continuum growth-fragmentation embedding.
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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

2 major / 6 minor

Summary. The note establishes an explicit combinatorial identity relating the perimeter process of the percolation-interface peeling of a Dobrushin-percolated triangulation to its Kreweras walk under the Bernardi–Holden–Sun bijection. Theorem 1 states that if T(n) are the spine-step times of the walk X ending at X(N), then the interface perimeter satisfies P(n) = ||X(T(n+1)) − X(N)||_1 + 2. Theorem 10 extends the same L1 formula to every branch of the branching peeling exploration, replacing the global endpoint by suitable restricted spine times T_t and closure times D_t. The argument proceeds by identifying reduced Kreweras words with peeling steps (Lemma 6) via the spine and the unfilled peeling process, then matching increments.

Significance. The result supplies a clean discrete counterpart to the embedding of a self-similar growth-fragmentation inside a Brownian cone excursion obtained by Da Silva–Powell–Watson, and thereby clarifies the link between the mating-of-trees and peeling approaches to percolated triangulations. The derivation is purely combinatorial, parameter-free, and makes essential but transparent use of the published BHS bijection; once the reduced-word/spine correspondence is granted, the perimeter identity follows by elementary increment matching. The branching extension (Theorem 10) is the load-bearing discrete input needed for a prospective joint scaling limit of Kreweras walks, exploration trees, and growth-fragmentations. As a short note it is appropriately scoped and adds a concrete geometric dictionary that was previously missing.

major comments (2)
  1. [§3.1, Lemma 8] Lemma 8 is the load-bearing bridge from the BHS bijection to the reduced-word/peeling identification, yet it is only asserted as “implied by the proof of Theorem 3.6 of [BHS23]”. For a note whose entire argument rests on this correspondence (close matchings ↔ monochromatic faces of the spine, ordered along the interface), a short self-contained extraction or a precise pointer to the relevant paragraphs/figures of BHS23 would make the claim checkable without reconstructing that proof. The subsequent induction in the proof of Lemma 6 and the recursive restriction in the proof of Theorem 10 inherit any gap here.
  2. [§4, proof of Theorem 10] In the off-interface case of the proof of Theorem 10, after the first separating L_ℓ/R_ℓ step the argument reduces to Φ(w_{T(j_0)+1}⋯w_q) and then says “we recursively iterate the same reasoning”. It should be made explicit that each recursive call updates the ambient walk to the corresponding cone-excursion subword and that the closure times D_t(n) for later spine steps are exactly the successive close-matching c-steps of those nested excursions (so that the L1 formula continues to hold with the correct local endpoint). The current wording leaves the identification of D_t slightly implicit.
minor comments (6)
  1. [§1 and §4] Theorem 2 is labelled “informal” and then restated as Theorem 10; consider merging into a single numbered statement (with the informal version only in the introduction) to avoid dual numbering.
  2. [§1, before Theorem 1] In the definition of spine steps just above Theorem 1, “cone excursion” is defined via X(m) ∈ X(n_1) + N²; the same notion reappears later with “height of the cone excursion”. A single consistent terminology (and a pointer that this matches the close-matching language of §3) would help.
  3. [Figure 1] Figure 1 caption and the bottom panel are helpful; adding a short legend that the red squares are exactly the spine times T(n) would make the visual claim of the theorem immediate.
  4. [Figure 1 caption] Typo: “T op left” / “T op right” in the Figure 1 caption (stray space after T).
  5. [Remark 3] Remark 3 correctly notes that the one-white-boundary restriction is presentational; a one-sentence indication of how the L1 formula changes for general (ℓ,r) Dobrushin data (first coordinate = white boundary vertices of the unexplored region, etc.) would make the remark immediately usable.
  6. [Title page] The date on the title page reads “July 31, 2026”; confirm this is intentional.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: external bijection plus independent increment matching yields the L1 identity

full rationale

The note takes the published Bernardi–Holden–Sun bijection Φ (and the reduced-word/spine correspondence implied by BHS23 Thm 3.6) as an external black box, then derives Theorems 1 and 10 by an elementary geometric calculation: the walk Y built from the reduced Kreweras word has the same increments as the perimeter process P, and the same initial value, so P(n) = ||X(T(n+1))−X(N)||_1 + 2. The target L1 formula is not assumed, fitted, or smuggled in via self-citation; spine steps and perimeter are independently defined objects whose equality is proved by matching. Branching (Thm 10) is the same identity applied recursively to restricted walks. Authors do not cite their own prior uniqueness theorems. Score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The paper is a pure combinatorial derivation resting on one external bijective correspondence and standard definitions of peeling and Kreweras walks. No parameters are fitted; the only non-standard ingredients are the spine / reduced-word notions already present in BHS23 and the branching-exploration bookkeeping introduced for the statement of Theorem 10.

assumptions (3)
  • domain assumption Bernardi–Holden–Sun bijection Φ: Kreweras walks in the quadrant are in bijection with Dobrushin-percolated triangulations (BHS23 Thm 2.11, Cor 2.13).
    Invoked throughout Sections 2–4 as the encoding that lets spine steps of the walk be compared with peeling steps of the map.
  • domain assumption Close-matchings in the Kreweras word correspond to monochromatic faces of the spine, and spine triangles appear in interface order (BHS23 Thm 3.6).
    Lemma 8 is stated as ‘implied by the proof of’ that theorem; the whole identification of reduced words with peeling steps rests on it.
  • standard math Standard definition of the (unfilled / branching) peeling process along a percolation interface and the resulting perimeter process (Angel–Curien, Budd, etc.).
    Used to define P(n) and P_t(n); no novelty claimed.
invented entities (2)
  • Spine steps / reduced Kreweras word (as used for perimeter extraction) independent evidence
    purpose: Select the subsequence of the walk whose L1 distances recover the perimeter process.
    The notions already appear in BHS23; the paper only re-purposes them to read off perimeters. No new continuum or physical entity is postulated.
  • Closure times D_t(n) for branching branches independent evidence
    purpose: Mark the end of the cone excursion that contains a given side-branch so that the L1 formula still holds off the main interface.
    Defined ad hoc in Section 4 for the statement of Theorem 10; they are combinatorial bookkeeping devices, not new physical objects.

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Pith. "Pith review of From Kreweras walks to branching perimeter processes of percolated triangulations." pith.science (2026). https://pith.science/paper/NYUFR3PK

@misc{pith2026260728566,
  author       = {Pith},
  title        = {Pith review of: From Kreweras walks to branching perimeter processes of percolated triangulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NYUFR3PK}},
  note         = {Machine review of arXiv:2607.28566}
}
read the original abstract

In this note, using a result of Bernardi, Holden and Sun, we give an explicit geometric relation between the perimeter of the peeling process along the percolation interface of a triangulation, and the corresponding Kreweras walk. The relation naturally extends to the branching peeling exploration. This sheds light on the relation between the growth-fragmentation process and correlated Brownian excursions discovered by Da Silva, Powell and Watson.

Figures

Figures reproduced from arXiv: 2607.28566 by the authors.

Figure 1
Figure 1. Top left: a Kreweras walk with 200 steps. Top right: the equivalent percolated triangulation, along with its percolation interface. Bottom: the length of the perimeter when exploring along the interface. The top equivalence and bottom-right implication are known; in this note, we show that the perimeter curve is obtained from the Kreweras walk by the L1 distance of certain non-excursion points (marked by red squares… view at source ↗
Figure 2
Figure 2. The four types of exploration steps Ca, Cb, Lℓ, Rℓ in the peeling process. The peeled edge corre￾sponds to the white-black edge on the bottom. In each step, a triangle is discovered and in every step of the form Lℓ, Rℓ, a submap with perimeter ℓ + 1 is filled in as “explored”. In each case, the new peeled edge is the new edge with bicoloured endpoints [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. The perimeter of the peeling process along the percolation interface. The edges on the boundary [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Building the map for the bijection, step by step. Red crosses indicate inactive edges. The top [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: The relation between the objects involved in the proof of Lemma [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Left: a percolated triangulation (M, σ), together with its percolation interface. Right: The spine Spine(M, σ). The monochromatic faces have been coloured accordingly. Double edges are shown in red; all boundary edges except the top are inactive. While the spine is obt…
Figure 7
Figure 7. Figure 7: The Kreweras walk and the values of Tt, Dt and T for the 11-letter word aaaaccaccbc. The reduced word is aa¯4 ¯b1. The triangle t which gives mt = 7 is not on the interface path, so the reduced word of X|[[0,mt]] is not a prefix of the reduced word of the entire walk, …
Figure 8
Figure 8. Figure 8: Top left: a Kreweras walk with 200 steps. Top right: the equivalent percolated triangulation, along with its exploration tree. When a branch breaks off from its parent, it is drawn in a different colour. Each branch ends at a different triangle t. Bottom: the length of…

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Reviewed July 31, 2026 · model on record in the stance chip above.