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In high-genus random triangulations the distance between random points converges in probability to a constant times log n.

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2026-06-26 02:33 UTC pith:RAKLH2B4

load-bearing objection The paper resolves the Budzinski-Chapuy-Louf conjecture by proving convergence in probability of rescaled distances to a constant via local convergence and isoperimetric bounds on ball growth.

arxiv 2606.27357 v1 pith:RAKLH2B4 submitted 2026-06-25 math.PR math.CO

Typical distances in high-genus triangulations

Tanguy Lions This is my paper
classification math.PR math.CO
keywords random triangulationshigh genusgraph distanceconvergence in probabilityvolume growthisoperimetric inequalities
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that when the genus of a uniform random triangulation grows linearly with the number of faces, the graph distance between two uniformly chosen points divided by log n converges in probability to a deterministic constant. This resolves an earlier conjecture and shows that distances are governed by the exponential volume growth of metric balls at logarithmic scales. The argument rests on controlling the size of those balls through local convergence theorems for triangulations with boundaries together with isoperimetric inequalities. A reader would conclude that the global metric of the surface is then determined by this local growth rate.

Core claim

We show that the distance rescaled by log(n) converges in probability to a deterministic constant. The proof relies on the precise study of the volume growth of the ball of radius r for r of order log(n). The main ingredients are the recent local convergence results for uniform triangulations with boundaries and the isoperimetric inequalities obtained by Budzinski and Louf.

What carries the argument

Volume growth of balls of radius r of order log n, controlled by local convergence of triangulations with boundaries and isoperimetric inequalities.

Load-bearing premise

Local convergence results for uniform triangulations with boundaries together with isoperimetric inequalities suffice to control the volume growth of balls of radius order log n in the high-genus regime.

What would settle it

Numerical sampling of many high-genus triangulations at large n showing that distance divided by log n fails to concentrate around any single value would falsify the claimed convergence in probability.

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If this is right

  • The typical distance is asymptotic to c log n for an explicit deterministic constant c.
  • The conjecture of Budzinski, Chapuy and Louf on high-genus distances is settled.
  • Global distances are governed by the local volume growth at logarithmic radii.
  • High-genus triangulations exhibit the same logarithmic distance scaling as hyperbolic surfaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scaling may hold for other families of random maps with genus linear in the size.
  • The deterministic constant could be computed explicitly from the local limit and the isoperimetric profile.
  • The result suggests that distances in random high-genus surfaces concentrate even when the surface is not triangulated.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper proves that in a uniform random triangulation with 2n faces and genus g proportional to n, the graph distance between two uniformly random vertices, rescaled by log n, converges in probability to a deterministic constant. This resolves a conjecture of Budzinski, Chapuy and Louf. The argument proceeds by establishing precise exponential volume growth for metric balls of radius r ~ log n (with high probability), using local convergence theorems for uniform triangulations with boundaries together with the Budzinski-Louf isoperimetric inequalities to control the growth rate up to the scale at which the ball exhausts the map.

Significance. If the result holds, it extends the theory of typical distances in random planar maps to the high-genus regime (g ∝ n), where global topology is dense, and shows that local geometric controls still produce deterministic macroscopic distances. The manuscript explicitly builds on the cited local-convergence and isoperimetric results without introducing free parameters, which is a methodological strength. The stress-test concern about genus-induced corrections does not land: the use of boundary local limits is precisely the tool that decouples local volume growth from global topology, and the isoperimetric bounds close the error terms uniformly enough for the required concentration.

minor comments (3)
  1. Introduction, paragraph following the statement of the main result: the deterministic constant is described only as 'the solution to an implicit equation'; an explicit variational characterization or reference to the growth rate α appearing later in the volume-growth analysis would improve readability.
  2. Section 2.2 (local convergence setup): the notation for the boundary condition in the local limit (e.g., the perimeter parameter) is introduced without a forward reference to how it is chosen when r ~ log n; a single clarifying sentence would prevent the reader from having to backtrack.
  3. The bibliography entry for Budzinski-Louf isoperimetric inequalities should include the precise theorem number used in the volume-growth argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that it resolves the conjecture of Budzinski, Chapuy and Louf via local convergence and isoperimetric controls. The recommendation is for minor revision, but no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity; derivation rests on external cited results for local convergence and isoperimetric control

full rationale

The paper's central argument studies volume growth of balls of radius r ~ log n via local convergence of uniform triangulations with boundaries together with Budzinski-Louf isoperimetric inequalities, then deduces that rescaled distances converge in probability to a constant. These ingredients are cited from prior work by different authors (Budzinski, Louf, Chapuy) and are not self-citations or self-definitions within the present manuscript. No parameter is fitted to a subset and then renamed a prediction, no ansatz is smuggled via the author's own prior work, and no uniqueness theorem is imported from the same authors. The derivation chain therefore remains self-contained against external benchmarks rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result depends on the validity and applicability of these previously established results to the proportional genus case.

axioms (2)
  • domain assumption Local convergence results for uniform triangulations with boundaries
    Cited as a main ingredient for studying volume growth
  • domain assumption Isoperimetric inequalities by Budzinski and Louf
    Used to control the growth of balls in the triangulation

reviewed 2026-06-26 · how reviews work

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Cite this review

Pith. "Pith review of Typical distances in high-genus triangulations." pith.science (2026). https://pith.science/paper/RAKLH2B4

@misc{pith2026260627357,
  author       = {Pith},
  title        = {Pith review of: Typical distances in high-genus triangulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RAKLH2B4}},
  note         = {Machine review of arXiv:2606.27357}
}
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read the original abstract

We study the distance between two uniformly chosen points on a uniform random triangulation whose genus g is proportional to the number of faces 2n. We show that the distance rescaled by log(n) converges in probability to a deterministic constant, which answers a conjecture of Budzinski, Chapuy and Louf. The proof relies on the precise study of the volume growth of the ball of radius r for r of order log(n). The main ingredients are the recent local convergence results for uniform triangulations with boundaries and the isoperimetric inequalities obtained by Budzinski and Louf.

Figures

Figures reproduced from arXiv: 2606.27357 by Tanguy Lions.

Figure 1
Figure 1. Figure 1: On the left, we represent the ball of radius [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: In these three figures, we assume that |Br(Tn,gn , xn)| = o(n) and |Br(Tn,gn , xn)| is of order at least log(n). On the left, we represent a pathological situation where both T 2 and T 3 contain a positive proportion of the total mass of Tn,gn . By the isoperimetric inequality, this event occurs with small probability. In the middle, the component T 3 contains almost all the mass of Tn,gn . This is the mos… view at source ↗
Figure 3
Figure 3. Figure 3: Left: a triangulation of the (6, 8, 9, 14)-gon of genus 5. Right: a triangulation with holes t0 ⊂ t, which has 1 boundary and 5 holes. Triangulations of multi-polygons t1 and t2 are the connected components of t \ t0. The triangulation t1 (resp. t2) is a triangulation of the (6, 8, 5, 13)-gon (resp. (14, 2, 4, 7)-gon). For p = (p1, · · · , pℓ) ∈ (N ∗ ) ℓ , we denote by Tp(n, g) the set of triangulations of… view at source ↗
Figure 4
Figure 4. Figure 4: We represent a triangulation t of the p-gon. On the left, we represent the strong balls of radius 2 centered at e1 (in red), e2 (in blue), and e3 (in green). Note that B + 2 (t, e1) contains the boundary on which e1 lies. Moreover, in B + 1 (t, e3), one of the green internal faces shares a vertex with a boundary face, and thus, this boundary face also belongs to B + 2 (t, e3). On the right, we represent in… view at source ↗
Figure 5
Figure 5. Figure 5: On the left: a realisation of Hλ. On the right, a triangulation t with one infinite boundary (dark grey face) and one infinite hole (white face) such that t ⊂ Hλ. On this example, |t|in = 17 denotes the number of green vertices and |∂ ∗ t| − |∂t| = 8 − 5 = 3 denotes the number of red edges minus the number of blue edges. For any 0 < λ ≤ λc, let h(λ) be the unique h ∈ (0, 1 4 ] such that λ = h (1+8h) 3/2 , … view at source ↗
Figure 6
Figure 6. Figure 6: On the left, a triangulation t of the p-gon. In the middle, we have B+ r (t, ∂t) = (t1, t3) and t \ B+ r (t, ∂t) = (t2, t4). On the right, we represent the hull of radius r, for which B• r (t, ∂t) = (t1, t3 ∪ t4) and t \ B• r (t, ∂t) = t2. In the remainder of the paper, the key task is to understand the volume growth of strong balls, that is, the quantity |B+ r (Tn,gn , en)|, where en is an oriented edge c… view at source ↗
Figure 7
Figure 7. Figure 7: We represent a triangulation of the p-gon t, the hull B• 1 (t, ∂t) in light grey, a segment I of length 2a + 1 lying on ∂it and B• 1 (t, I) in orange. On this example, B• 1 (t, I) ∩ ∂ ∗B• 1 (t, ∂t) = ∅, i.e. the blue segment does not lie on the hole of B• 1 (t, ∂t). Let us describe how Section 4.2 is organised: • In Section 4.2.1, we use Theorem 2.13 to show that (Tn,gn,pn , en ) typically has (locally) th… view at source ↗
Figure 8
Figure 8. Figure 8: On this example, we represent a triangulation [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: On the left: we illustrate the segment ◦ Ja(t, e) obtained by removing the a 3 4 right￾most and leftmost edges from Ja(t, e). On the right: we represent a triangulation (t, e) that satisfies GOOD(a, δ) and illustrate the different points. For any a, δ > 0, we introduce Bn(a, δ) as the event on which (Tn,gn,pn , en ) satisfies GOOD−(a, δ). We define Cn(a, δ) as the event on which (Tn,gn,pn , en ) satisfies … view at source ↗
Figure 10
Figure 10. Figure 10: The triangulation with holes tk for k ∈ {0, . . . , τa − 1}. The variable Xk counts edges exposed on the hole that do not belong to the boundary, while Yk counts boundary edges that have been swallowed. Since Ja is almost surely a segment, it suffices to show P  Xτa ∈ [(1 − δ)m−1 θ 2a,(1 + δ)m−1 θ 2a]  ≥ 1 − η 2 . Let Sk = Xk − Yk denote the evolution of the perimeter process. Its increments ∆Sk := Sk+1… view at source ↗
Figure 11
Figure 11. Figure 11: In this example, e ′ ∈ Ja(t, e). Hence ψ(e ′ ) ∈ Ia(t, e), or equivalently e ∈ Ia(t, ψ(e ′ )). For e ∈ ∂t, we define X(e) := Ja(t, e) ∩ ∂ ∗B• 1 (t, ∂t). Using (25), we see that for each e ′ ∈ ∂ ∗B• 1 (t, ∂t), the edge e ′ belongs exactly to the sets X(e) for which e ∈ Ia(t, ψ(e ′ )). Since |Ia(t, ψ(e ′ ))| ≤ 2a + 1, we obtain X e∈∂t |X(e)| 2a + 1 ≤ |∂ ∗B • 1 (t, ∂t)| ≤ X e∈∂t |Ja(t, e)| |Ia(t, e)| , (26) … view at source ↗

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This paper was first reviewed by grok-4.3 on June 26, 2026.