Scaling limit of first passage percolation geodesics on planar maps

Emmanuel Kammerer

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Scaling limit of first passage percolation geodesics on planar maps

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Pith's one-line read This paper proves that the tree of first-passage percolation geodesics to the root in random Boltzmann planar maps of type $a\in(3/2,5/2)$ converges, after rescaling, to an explicit random tree constructed from a coalescing flow of pure…

desk verdict The time-reversal argument is a real advance, but the proof of the geodesic-tree scaling limit misses the step from fixed Ulam labels to degree-ranked faces. read the letter →

arxiv 2412.02666 v3 pith:F6SJOUYH submitted 2024-12-03 math.PR

Emmanuel Kammerer This is my paper · ORCID

classification math.PR MSC 60K3560F1760G5160J6005C80

keywords first-passagepercolationrandomplanarmapspeelingexplorationgeodesictreescoalescingflowgrowth-fragmentationscalinglimitsself-similarMarkovprocesses

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 paper proves a scaling limit for the collection of shortest paths to the root in first-passage percolation (FPP) on random Boltzmann planar maps. For weight sequences $q$ of type $a\in(3/2,5/2]$, it first shows that the number of faces met by a geodesic has explicit asymptotics: polynomial in the dilute regime $a>2$, logarithmic at $a=2$, and again logarithmic with a different rate in the dense regime $a<2$. Extending this control to all geodesics lets the author compare FPP balls with balls for the dual graph distance and, in the dense regime, bound the dual-graph diameter by $C_q\log\ell$. The main result is that, for $a\in(3/2,5/2)$, the tree of FPP geodesics to the root, rescaled by $\ell^{2-a}$, converges to an explicit random countable metric space built from a stochastic coalescing flow of pure jump diffusions, with distances read from Lamperti-transformed birth times in a growth-fragmentation cell system. The same flow is conjectured to underlie the scaling limit of the maps themselves when high-degree faces are present.

What carries the argument

The load-bearing mechanism is a time-reversal of the uniform peeling exploration, a step-by-step discovery of the map by revealing the face behind a uniformly chosen boundary edge. Proposition 2.1 states that, conditionally on the perimeter process, the backward-built sequence of maps has the same law as the forward explored regions, each with the explored region collapsed to a face. From this identity, a face discovered at a positive jump of the perimeter lies on the geodesic to the root independently, with probability $\theta_i=(2\Delta P(i)+1)/(2P(i+1))$ on $\Delta P(i)\ge 0$, giving the Bernoulli representation behind Theorem 1.1. To code the full tree of geodesics, the paper embeds the reversed exploration into a discrete coalescing flow on $\mathbb{R}/\mathbb{Z}$ whose trajectories coalesce when a face is glued; the continuum limit is the flow $X_t(v)=v+\int\int\int g(X_{s-}(v),z,u)\widetilde N(ds,dz,du)$, where $\widetilde N$ is a compensated Poisson point process of intensity $2 c_a p_q\, dt\,\lambda(dz)\,du$ and $\lambda$ is the Levy measure of the growth-fragmentation process. The function $g$ is a 1-periodic kernel satisfying a monotonicity condition, which is what allows existence and uniqueness of strong solutions by a non-Lipschitz SDE theorem. The metric space $T_a$ is then built from the tree of positive jumps of the growth-fragmentation, using the flow to decide coalescence times and the Lamperti transform to convert birth times into distances.

What would settle it

Simulate a finite Boltzmann map for a fixed type $a\in(3/2,5/2)$, compute the FPP geodesic tree from the root, and compare the empirical law of $\ell^{2-a} d_{T(m)}(f_i,f_j)$ for the largest-degree faces with the law of $d_{T_a}/(2 c_a p_q)$; a systematic disagreement for any $a$ refutes Theorem 1.5. A sharper local test is to condition on the perimeter process and check the Bernoulli property (2.5): the indicators that faces discovered at positive jumps lie on the geodesic should be independent with parameters $(2\Delta P(i)+1)/(2P(i+1))$, so any violation of independence or of those probabilities would falsify the time-reversal identity on which the paper rests.

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Extended reading notes

Core claim

The central discovery is a complete continuum description of the FPP geodesics to the root. Under the law $P^{(\ell)}$ on bipartite Boltzmann maps of perimeter $2\ell$, enumerate the faces $f_i$ in non-increasing degree order. Theorem 1.5 asserts that $(\ell^{2-a} d_{T(m)}(f_i,f_j))_{i,j}$ converges in distribution, in the product topology, to $((2 c_a p_q)^{-1} d_{T_a}(w_i,w_j))_{i,j}$, where $c_a=\pi/\Gamma(a)$, $p_q$ is the partition-function constant from (1.1), and $T_a$ is a random countable metric space defined through coalescing diffusions with jumps. The coalescence events building $T_a$ are exactly the positive jumps of the self-similar growth-fragmentation that arises as the scaling limit of the peeling perimeter; for $a<2$ a single large jump is responsible, while for $a\ge 2$ the coalescence is produced by an infinite accumulation of small jumps. The distance $d_{T_a}$ is obtained by applying the Lamperti transform to the birth times in the cell system and then taking the usual tree distance to the nearest common ancestor. The paper also proves the face-count asymptotics of Theorem 1.1, the inclusions of Corollaries 1.2 and 1.3, and the dense-phase diameter bound of Theorem 1.4; the tree limit itself is stated for $a\in(3/2,5/2)$, and the special case $a=5/2$ is left for future work because the absence of high-degree faces changes the form of the tree statement.

Load-bearing premise

The load-bearing premise is the exact distributional identity between the time-reversed backward construction and the forward peeling exploration conditional on the perimeter process (Proposition 2.1); if that identity is only approximate, the Bernoulli representation, the discrete coalescing flow, and ultimately the scaling limit of the geodesic tree lose their foundation.

Editorial extensions

If this is right

Theorem 1.1 gives explicit asymptotics for the number of faces on an FPP geodesic: in the dilute regime $a\in(2,5/2]$ the count, rescaled by $n^{-(a-2)/(a-1)}$, converges to $(e_q/2)\int_0^t ds/\Upsilon^\uparrow_a(p_q s)$, while at $a=2$ and in the dense regime $a\in(3/2,2)$ the logarithmic rates are $1/\pi^2$ and $(2/\pi)\tan((2-a)\pi)$, respectively.
Corollaries 1.2 and 1.3 locate FPP balls inside dual-graph balls with explicit constants: asymptotically $\mathrm{Ball}^{fpp}_r \subset \mathrm{Ball}^\dagger_{\lfloor(1+\varepsilon)e_q r\rfloor}$ in the dilute case, and with radius of order $r^2$ at $a=2$, improving earlier inclusions that had non-explicit constants.
Theorem 1.4 proves that in the dense phase $a\in(3/2,2)$ the diameter of a large Boltzmann map for the dual graph distance is $O(\log\ell)$ with high probability, matching the order expected when hub faces dominate the geometry.
Theorem 1.5 provides a countable but explicit continuum tree $T_a$ whose distances between the images of the largest-degree faces are the scaling limit of FPP tree distances; the coalescence structure is governed by positive jumps of the growth-fragmentation, with a qualitative change between $a<2$ and $a\ge 2$.
The convergence of the discrete coalescing flow holds also at $a=5/2$, giving a flow-based description of geodesics to the root in the Brownian disk, even though the tree statement of Theorem 1.5 needs a different formulation there.

Reading between the lines

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

If Theorem 1.5 holds, the same flow construction is a natural candidate for the scaling limit of the full metric space of high-degree maps; Section 6's shortcut-modified distance $d_{D_a}$ gives an explicit conjecture that FPP distances and, for $a\in(2,5/2)$, dual-graph distances converge to $d_{D_a}$ with a known multiplicative constant.
The Bernoulli representation suggests a direct numerical check that is much cheaper than simulating full continuum limits: condition on the perimeter of a finite map, record which discovered faces lie on the root geodesic, and test whether the indicators are independent with the stated probabilities; a mismatch would indicate the time-reversal identity is not exact.
The dichotomy between one large-jump coalescence for $a<2$ and accumulated small-jump coalescence for $a\ge 2$ is likely to appear in other observables of high-degree maps, such as the correlation decay of the largest faces or the mixing time of random walks on the dual map.
The paper leaves open the identification of the limiting flow at $a=5/2$ inside existing Brownian-disk constructions; proving that identification would give a new, flow-based route to geodesics in the Brownian disk.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know: this is a strong paper, but Theorem 1.5 as stated has a gap. The proof shows convergence for every fixed Ulam label w, then asserts that the degree-ranked faces converge to the same limit. The transfer is not proved.

What is actually new: the time reversal of the uniform peeling exploration is a clever tool, and it gives a clean Bernoulli representation for geodesic membership (eq. 2.5). The convergence of face counts along geodesics (Theorem 1.1) and the applications to ball inclusion and the logarithmic diameter in the dense phase (Corollaries 1.2 and 1.3, Theorem 1.4) look well supported. The coalescing flow of pure jump diffusions is a genuinely new object, and building the metric space T_a from it is a nice idea. The proofs in Sections 3 and 4 are detailed and check the necessary hypotheses, e.g., strong existence via [LP12].

The soft spot is in Section 5.3. Theorem 1.5 is about faces ordered by degree, but the convergence is established for fixed Ulam labels w. The degrees deg(f_w)/ℓ converge to Δ_w, and the distances for each fixed w,w′ converge. But the i-th largest face in the discrete map has a label w_i(ℓ) that is random and depends on ℓ. The paper never shows that w_i(ℓ) converges to the label of the i-th largest jump Δ_w. The citation to [CCM20] only gives no ties in the limit, not the discrete-to-continuum transfer of order statistics. So the literal statement of Theorem 1.5 does not follow from the proof. This is a proof gap, not a counterexample, and it may be fixable with a tightness argument or a more careful coupling, but it is load-bearing.

Everything else seems solid. The conjecture section is clearly separated from the proven results. The paper is long and delegates some steps to prior work, but that is normal in this area.

Who is this for? People working on random planar maps, peeling, and growth-fragmentations. It deserves a serious referee. I would send it out, with a note asking the referee to check whether the label-to-order transfer can be supplied. If that gap can be closed, this becomes an important paper.

Referee Report

2 major / 3 minor

Summary. The paper studies first-passage percolation (fpp) on the duals of random Boltzmann planar maps of type a in (3/2,5/2]. It has two main strands. First, it establishes scaling limits for the number of faces along fpp geodesics to the root, derives comparisons between fpp balls and dual-graph balls in the dilute phase, and proves a logarithmic upper bound for the dual-graph diameter in the dense phase. Second, it introduces a discrete coalescing flow encoding the fpp geodesics to the root, proves that the flow converges to a stochastic coalescing flow of pure jump diffusions driven by a Poisson point process, and uses the latter flow to define a countable random metric space T_a. Theorem 1.5 states that the rescaled tree of fpp geodesics, indexed by degree-ranked faces, converges to T_a. Section 6 constructs conjectural metric spaces D_a obtained by adding shortcuts to T_a.

Significance. If the main theorem is fully established, this is a substantial advance: it gives an explicit continuum description of the fpp geodesic tree in a family of random planar maps that is outside the Brownian case, and it ties this description to the known growth-fragmentation scaling of perimeters. The time-reversal construction in Section 2 is elegant, and the proof of Proposition 2.1 by induction is convincing. The proof of the flow convergence in Section 4 is detailed, with careful verification of the hypotheses of [LP12] and a systematic truncation argument for small jumps. The applications in Section 3 give explicit constants and improve earlier non-explicit comparisons. The main weakness is a missing transfer from fixed Ulam labels to degree-ranked faces in the proof of Theorem 1.5; the fixed-label convergence is proved, but the literal statement of the theorem requires an additional reordering argument.

major comments (2)

[Section 5.3, Eq. (5.8)] The proof of Theorem 1.5 establishes convergence for fixed Ulam labels w, w' in U, and it proves deg(f_w^{(l)})/l converges to Delta_w. However, Theorem 1.5 is stated for the faces (f_i) arranged in non-increasing order of degree. The random Ulam label w_i^{(l)} of the i-th largest-degree face depends on l and is random, and the proof never shows that w_i^{(l)} converges to the label w_i of the i-th largest positive jump Delta_w in the continuum cell system. The sentence invoking Proposition 3 of [CCM20] addresses uniqueness and ordering of the continuum jumps, not the discrete-to-continuum label transfer under the joint convergence of degrees and distances. A transfer lemma is needed; without it the fixed-label convergence does not imply the product-topology convergence of the distance matrix indexed by degree-ranked faces. The statement also needs a tie-breaking convention for equal-degree faces, or a proof that ties are asymptotically negligible, since the discrete degrees are integer-valued.
[Section 5.3, a in [2,5/2) case] In the case a in [2,5/2), the proof of the limsup inequality uses a sequence (z i_n) in J_a and the corresponding discrete faces f_{z i_n}^{(l)} as approximate common ancestors, then lets epsilon go to 0. This argument is plausible, but it relies on the same unproved reordering transfer: the discrete ancestors must be shown to correspond to the faces that appear with the labels used in Theorem 1.5 after degree-ordering. The liminf inequality is justified only for fixed labels. The convergence of the discrete coalescence times and ancestors to the continuum coalescence time, including the case where the continuum common ancestor is an element of J_a, should be stated and proved as part of the missing transfer lemma.

minor comments (3)

[Section 1.3] The section title 'Perpectives' contains a typo and should read 'Perspectives'.
[Section 3.3, proof of Corollary 1.2] The phrase 'by a union bound one,' is grammatically incomplete; the sentence should be restructured, for example by replacing it with 'by a union bound,' and continuing with the probability estimate.
[Section 1.1] In the paragraph on the critical non-generic case, the sentence 'the scaling limit of random maps of law P^{(1)} conditioned to have n vertices... equipped with de distanced_gr' contains a garbled phrase 'de distanced_gr'; it should read 'equipped with the distance d_gr'.

Circularity Check

0 steps flagged · score 2.0 of 10

No meaningful circularity: the main geodesic scaling limit is derived from an exact time-reversal and external perimeter scaling limits; self-citations are auxiliary.

full rationale

The central derivation is self-contained rather than circular. Theorem 1.5's continuum limit T_a is built from the growth-fragmentation cell system and a flow of coalescing jump diffusions whose jump measure and Lamperti time changes are expressed in terms of the model constants p_q, c_a, and e_q derived from the weight sequence and partition-function asymptotics, not fitted to geodesic data. The discrete coalescing flow is constructed from the time-reversal of the uniform peeling exploration, and its convergence (Theorem 4.4) is established by a genuine coupling argument with small jumps removed, ultimately relying on standard martingale and weak-convergence criteria. The distributional identity in Proposition 2.1 is proven by induction, so the Bernoulli representation of geodesic membership is an exact consequence rather than an assumed input. The limit distance d_Ta is read from Lamperti-transformed birth times in the cell system; the convergence of the corresponding discrete quantities follows from the jointly established scaling limits of the perimeter processes and jump times, not from postulating the limit. Self-citations to [Kam23] and [Kam25] appear in the proof of Theorem 1.4 and in Remark 3.4 for martingale tools and a constant evaluation, but these are published results with stated assumptions independent of the present target theorem, and they are not used to force the main geodesic-tree convergence. The skeptic's concern that Theorem 1.5 is proved for fixed Ulam labels rather than for degree-ranked faces identifies a potential proof gap in the transfer from label convergence to order-statistic convergence, but this is a missing uniformity argument, not a circular reduction of the claimed prediction to its inputs. No equation or fitted parameter is reused as its own prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 3 invented entities

The theorems are derived from a large network of established peeling and growth-fragmentation results; no numbers are fitted to data. The new limit objects, namely the stochastic flow, T_a, and D_a, are introduced within the paper, with T_a proven to be the limit and D_a conjectural.

assumptions (4)

standard math Scaling limit of the perimeter process under P^(1)_∞: (P∞(nt)/n^(1/(a-1)))_{t≥0} converges to (Υ↑_a(p_q t))_{t≥0} (Proposition 10.3 of [Cur23]).
Used in (3.1) to reduce Theorem 1.1 to continuous-time estimates for the stable Lévy process conditioned to stay positive.
standard math Under P^(ℓ), the rescaled perimeter process converges to the positive self-similar Markov process X_a^(1-a) (Proposition 6.6 of [BBCK18]), giving (4.2), (4.3) and (4.4).
This is the backbone of the continuous-time scaling in Section 4 and of the flow limit in Theorem 4.4.
domain assumption Spatial Markov property and branching Markov property of peeling explorations (Propositions 4.6 and 6.3 of [Cur23]).
Used in Section 5.2 to build the branching cell system of independent explorations inside holes, from which the geodesic tree is reconstructed.
standard math Strong existence and uniqueness for the jump SDE (4.7) follows from Theorem 5.1 of [LP12] after verifying monotonicity and growth conditions in Lemma 4.1.
The coalescing flow is defined through this SDE; the lemma checks the hypotheses (3.b) and (5.a) of [LP12].

invented entities (3)

Stochastic coalescing flow of pure jump diffusions (X_t(v))_{v,t} driven by the compensated PPP in (4.7)
purpose: Continuum limit of the discrete coalescing flow encoding fpp geodesics; trajectories coalesce on R/Z and carry the genealogy of geodesics.
Proved in Theorem 4.4 to be the scaling limit of the discrete flow, but all evidence is internal to this paper; no independent external prediction is given.
Random countable metric space T_a (Section 5.1), whose points are positive jumps of the growth-fragmentation plus the root, with distance via Lamperti-transformed birth times.
purpose: Claimed and proven scaling limit of the tree of fpp geodesics to the root.
Theorem 1.5 establishes convergence internally; there is no independent experimental or external mathematical verification.
Random metric space D_a with shortcuts built from negative jumps (Section 6), conjectured scaling limit of high-degree maps under fpp or dual graph distances.
purpose: Conjectural answer for the scaling limit of the full map equipped with fpp or dual graph distance.
Explicitly conjectured in Conjectures 6.2 to 6.4; no proof or external support is claimed.

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Pith. "Pith review of Scaling limit of first passage percolation geodesics on planar maps." pith.science (2026). https://pith.science/paper/F6SJOUYH

@misc{pith2026241202666,
  author       = {Pith},
  title        = {Pith review of: Scaling limit of first passage percolation geodesics on planar maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F6SJOUYH}},
  note         = {Machine review of arXiv:2412.02666}
}

read the original abstract

We establish the scaling limit of the geodesics to the root for the first passage percolation distance on random planar maps. We first describe the scaling limit of the number of faces along the geodesics. This result enables us to compare the metric balls for the first passage percolation and the dual graph distance. It also enables us to give an upper bound for the diameter of large random maps. Then, we describe the scaling limit of the tree of first passage percolation geodesics to the root via a stochastic coalescing flow of pure jump diffusions. Using this stochastic flow, we also construct some random metric spaces which we conjecture to be the scaling limits of random planar maps with high degrees. The main tool in this work is a time-reversal of the uniform peeling exploration.

Figures

Figures reproduced from arXiv: 2412.02666 by the authors.

**Figure 2.** Left: a planar map m ∈ M(4). The root edge is marked with an arrow and the root face corresponds to the outer face. Right: illustration for the tree T (m) in dashed green lines of fpp geodesics to the root face fr. Note that T (m) is not a deterministic function of m but depends on the random exponential edge lengths. en follows the growth of the fpp balls of center fr. 1.2 Main results Our results can be divided in… view at source ↗

**Figure 3.** The first five steps of the exploration of the map [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: The five steps of the reversed uniform exploration [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: The embedding of the reversed uniform exploration [PITH_FULL_IMAGE:figures/full_fig_p037_5.png]

**Figure 6.** Figure 6: The trajectory associated with a positive jump [PITH_FULL_IMAGE:figures/full_fig_p045_6.png]

**Figure 7.** Figure 7: Coalescence of the fpp geodesics from f1 and f2 to the root face fr. • If f = fr or f ′ = fr, then c (ℓ) (f, f′ ) = fr; • If f = f ′ , then c (ℓ) (f, f′ ) = f = f ′ ; • If f ̸= f ′ and f, f′ ̸= fr, then let z be the largest prefix of wi, w′ i ′ . Let ∅ = z0 ≺ z1 ≺ . . …

**Figure 8.** Figure 8: Sketch of the end of the proof of Theorem [PITH_FULL_IMAGE:figures/full_fig_p052_8.png]

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Forward citations

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

The scaling limit of planar maps with large faces
math.PR 2025-01 accept novelty 8.0 of 10

Large Boltzmann planar maps with a stable exponent alpha converge in the Gromov-Hausdorff-Prokhorov sense to an explicit random compact metric space S_alpha of Hausdorff dimension 2 alpha.

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David Applebaum. L \'e vy processes and stochastic calculus. , volume 116 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2nd ed. edition, 2009

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work page 2018

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