{"id":"8b302dad-90e8-4372-9231-a49ff6c42650","arxiv_id":"2412.02666","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The tree of first-passage percolation geodesics on random Boltzmann planar maps converges after scaling to a countable metric space driven by a coalescing flow of jump diffusions.","lead":"This paper proves that the fastest paths from every face to the root in random planar maps with randomly weighted edges converge, after rescaling, to a random tree built from a stochastic flow of coalescing jump diffusions. The same arguments give a logarithmic diameter bound and clarify how two natural notions of distance on these maps compare.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.5's proof proves convergence for Ulam-labelled faces, not for the degree-ranked faces (f_i) in the statement; the transfer from label convergence to degree-order convergence is asserted but not proved.","rationale":"I read the paper in good faith and found the time-reversal construction, Proposition 2.1, and the Bernoulli representation (2.5) plausible; the simple constant-perimeter examples that initially look like counterexamples actually satisfy the claimed independence, so I do not share the reader's weakest-assumption concern. The proofs in Sections 3 and 4 are detailed and appear internally consistent modulo several typographical-level issues (e.g., the definition of N^{(ℓ),T} in Section 4.3 is written with Δ eP(t)>0 while the surrounding arguments and Lemma 4.6 require all nonzero jumps). The main gap I see is at the very end of the proof of Theorem 1.5: the theorem is stated for faces ranked by degree, but the proof only establishes convergence for faces carrying fixed Ulam labels. No argument is given that the random label of the i-th largest-degree face stabilizes to the fixed label of the i-th largest continuum positive jump, nor that the product-topology distance matrix for the reordered faces follows from the labelled convergence. This is a missing step in the proof of the central claim, so I recommend a conditional acceptance: the theorem is very likely true, but the missing transfer lemma should be supplied before the paper is accepted as is.","tokens_in":51499,"tokens_out":40581,"duration_ms":395656,"concrete_test":"Write and prove a standalone transfer lemma: under the joint convergence in (5.8)-(5.11), for each i ≥ 1, the label w_i(ℓ) of the i-th largest-degree face converges in distribution to the label of the i-th largest positive jump Δ_w in the limiting cell system, and the corresponding distance matrix converges for the product topology. If the lemma fails, exhibit a sequence of labelled trees satisfying the fixed-label convergences for which the degree-ranked distance matrix does not converge; if it holds, insert the proof before the final sentence of Section 5.3.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central theorem states convergence of the matrix ℓ^{2-a} d_T(m)(f_i, f_j) where (f_i) are the faces in non-increasing order of degree. Section 5.3 proves, for fixed Ulam labels w,w' ∈ U, that ℓ^{2-a} d_T(m)(f_w^{(ℓ)}, f_{w'}^{(ℓ)}) converges to (1/(2 c_a p_q)) d_Ta(w,w'), and that normalized degrees deg(f_w^{(ℓ)})/ℓ converge to Δ_w. But the i-th largest-degree face in the discrete map has a label w_i(ℓ) that depends on ℓ and is random. The proof never shows that w_i(ℓ) converges to the label w_i of the i-th largest positive jump Δ_w in the limiting cell system, nor that the distance matrix indexed by these reordered random labels converges in the product topology. The sentence citing Proposition 3 of [CCM20] addresses uniqueness and ordering of the continuum jumps, but not the discrete-to-continuum label transfer under the joint convergence of degrees and distances. Without this missing transfer lemma, the fixed-label convergence does not imply the stated convergence for degree-ranked faces. This is a proof gap, not a demonstrated counterexample, but it is load-bearing because the theorem's literal statement is about degree ordering.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":51827,"tokens_out":5116,"duration_ms":58992,"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":[{"comment":"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":"Section 5.3, Eq. (5.8)"},{"comment":"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.","section":"Section 5.3, a in [2,5/2) case"}],"minor_comments":[{"comment":"The section title 'Perpectives' contains a typo and should read 'Perspectives'.","section":"Section 1.3"},{"comment":"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":"Section 3.3, proof of Corollary 1.2"},{"comment":"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'.","section":"Section 1.1"}],"recommendation":"major_revision","confidential_remarks":"I largely agree with the reader's assessment. The missing label-transfer lemma in the proof of Theorem 1.5 is a real and load-bearing gap, but it appears fixable within the manuscript's scope by adding a lemma that proves convergence of the degree-ranked labels and of the reordered distance matrix. I did not find a problem with Proposition 2.1; the induction in Section 2.3 is convincing. The paper is otherwise sound and well within the scope of the journal, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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].\n\nThe 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.\n\nEverything 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.\n\nWho 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.","headline":"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.","tokens_in":52385,"tokens_out":4139,"would_cite":true,"duration_ms":41842,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60F17","60G51","60J60","05C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["first-passage percolation","random planar maps","peeling exploration","geodesic trees","coalescing flow","growth-fragmentation","scaling limits","self-similar Markov processes"],"falsifier":"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.","tokens_in":51267,"feed_emoji":"🌳","tokens_out":9417,"duration_ms":82864,"temperature":0.7,"pith_summary":"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.","feed_headline":"FPP geodesic trees on planar maps get a continuum limit","feed_subtitle":"The limit is built from a flow of coalescing diffusions, with distances read off by a Lamperti transform.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the peeling construction, the spatial Markov property, and the law of the perimeter process on which Proposition 2.1 and all subsequent asymptotics rest.","marker":"[Cur23]"},{"why":"Defines self-similar growth-fragmentations and provides the scaling limit of the perimeter process, giving the cell system and Lamperti transform used in the continuum tree.","marker":"[BBCK18]"},{"why":"Establishes the stable-map geometry and the tail and local-limit estimates for the perimeter process used in the face-count asymptotics of Theorem 1.1.","marker":"[BC17]"},{"why":"Supplies the local limit, uniform integrability, and decorrelation lemmas for the perimeter process that drive the dilute, critical, and dense regimes of Theorem 1.1.","marker":"[BCM18]"},{"why":"Provides the martingales and re-rooting trick that turn face-count controls into the dense-phase diameter bound of Theorem 1.4.","marker":"[Kam25]"},{"why":"Gives the existence and uniqueness theorem for the jump SDE defining the continuum coalescing flow, which is not Lipschitz.","marker":"[LP12]"},{"why":"Provides the weak-convergence criteria for jump Markov processes used to prove convergence of the discrete coalescing flow to the continuum flow.","marker":"[JS87]"},{"why":"Gives the earlier inclusion between FPP and dual-graph balls with non-explicit constants that Corollary 1.2 improves by identifying the constant $e_q$.","marker":"[CM20]"}],"fun_headline_variants":["FPP geodesics on planar maps now have a scaling limit","Coalescing diffusions describe FPP geodesic trees","Planar map geodesics converge to a continuum tree","Scaling limit found for FPP distances on random maps","Tree of FPP geodesics to root has a continuum limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["FPP geodesics on planar maps now have a scaling limit","Coalescing diffusions describe FPP geodesic trees","Planar map geodesics converge to a continuum tree","Scaling limit found for FPP distances on random maps","Tree of FPP geodesics to root has a continuum limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000783,"raw_usage":{"total_tokens":3502,"prompt_tokens":1032,"completion_tokens":2470,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":648,"completion_tokens_details":{"reasoning_tokens":2398}},"tokens_in":648,"tokens_out":2470,"duration_ms":18185,"temperature":1.0,"reasoning_tokens":2398,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:12:50.079641+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}