{"id":"bec7c70f-9ebf-4433-83d8-2a8b53072ced","arxiv_id":"2505.05447","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The only rerooting-invariant Markovian random quadrangulations are Boltzmann maps, and every submap that induces a Markovian decomposition is a 'stopping map,' a class strictly larger than peeling explorations.","lead":"Random planar maps, made by gluing squares, often have a Markov property: if you know a patch of the map, the rest is independent and has the same type of law. This paper proves the only maps with this property and a rerooting symmetry are Boltzmann maps, and it classifies which random patches can be explored without breaking the property.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract's decorated characterization is false as stated: tree-supported rerooting-invariant Markov measures are counterexamples unless the non-degeneracy condition is added.","rationale":"The reader correctly identified the non-degeneracy assumption in Theorem 4.10 as the weakest point, and Remark 4.13 confirms that the abstract overstates the decorated characterization. I found no fatal error in the undecorated Theorem 3.20, and the stopping-map counterexample in Section 7 is an independent, concrete contribution that supports the paper's framework. The main concern is therefore not that the theorems are wrong but that the central umbrella claim, as stated in the abstract, is false without the caveat. This is a presentational and completeness issue rather than a refutation of the main technical results, so the existing CONDITIONAL verdict remains appropriate: the preprint should state the non-degeneracy hypothesis in the abstract and write out the deferred base-case arguments in Claim 4.12 and the corresponding metric claims before the characterization can be fully verified.","tokens_in":47299,"tokens_out":11094,"duration_ms":129558,"concrete_test":"Independently verify Claim 4.12 for ℓ = 2: derive p_{2,b}(t⊞_i, σ⊞) = q4 P_{2,b}(Q = t_i) from the Markov property and rerooting invariance, and prove q4 > 0 using only P_{2,b}(has a face) > 0 for all four shifted boundary conditions. Then check whether an arbitrary rerooting-invariant tree-supported family, for example the uniform measure over tree-type quadrangulations of each perimeter, satisfies Definition 4.9 and violates the Boltzmann conclusion. The abstract's claim is settled by this check.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing concern is the mismatch between the abstract's blanket characterization and the actual decorated theorem. Theorem 4.10 only characterizes rerooting-invariant Markov measures under the extra assumption P_{ℓ,b}(Q has at least one face) > 0, together with continuity in b. Remark 4.13 concedes that without non-degeneracy the equivalence fails: tree-supported measures satisfy the Markov property and rerooting invariance but need not be Boltzmann-decorated, because trees have no internal faces on which the spin interaction can act. This is not a cosmetic caveat. In the proof of (3)⇒(2), the base case of Claim 4.12 compares tree probabilities through an auxiliary four-face map ⊞, and the proof that the constant q4 is positive is exactly where the non-degeneracy hypothesis is used. The argument for q4 > 0 is compressed ('if q4 is 0 there have to be...') and is part of the text explicitly deferred to the reader. Since the abstract states the characterization without the caveat, the central umbrella claim is false as written; the theorem with the caveat may still be true.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a unifying framework for the spatial Markov property for quadrangulations with boundary, in three settings: undecorated maps, spin-decorated maps, and metric decorated maps. The main results are (i) a characterization, Theorems 1.2, 3.20, 4.10, and 5.25, of rerooting-invariant Markovian measures as Boltzmann-type measures with weight q^{|F|}, extended to spin decorations and to metric maps with an additional exponential edge weight; (ii) a characterization of random submaps that induce Markovian decompositions as 'stopping maps' for map-indexed filtrations (Theorems 1.3, 3.15, 6.2); and (iii) an example of a stopping map that cannot be obtained by a peeling algorithm (Section 7). The paper is largely self-contained, using map-indexed filtrations, monotone-class arguments, and Brownian-bridge decompositions in the metric setting.","tokens_in":47404,"tokens_out":7347,"duration_ms":84135,"significance":"If the stated characterizations are correct, they provide a clean converse to the well-known Markov property of Boltzmann planar maps: rerooting invariance plus the spatial Markov property forces the law to be Boltzmann, including decorated and metric analogues. The stopping-map framework is a useful contribution because it separates the strong Markov property from the algorithmic peeling procedure, and the metric extension with mid-edge stopping is a natural generalization motivated by the metric graph GFF. The paper also contains explicit, reproducible proof strategies: the ratio argument deriving q, the use of Lemma 2.2 for Brownian-bridge splitting, and the construction of a filtration from a local map in Theorem 6.2. However, the advertised umbrella claim for decorated maps is overbroad as stated, and several load-bearing steps are either explicitly deferred or not fully written, which currently prevents the results from being accepted in their advertised form.","major_comments":[{"comment":"The abstract states a characterization of rerooting-invariant Markovian laws as Boltzmann-type maps without any non-degeneracy hypothesis, but Theorem 4.10 requires both P_{ℓ,b}(Q has at least one face) > 0 and continuity in the boundary condition. Remark 4.13 explicitly concedes that without the non-degeneracy hypothesis the equivalence fails: tree-supported measures satisfy the Markov property and rerooting invariance but need not be Boltzmann-decorated, since trees carry no internal faces on which the spin interaction can act. This is not a cosmetic caveat: the abstract's blanket claim is false as written, and the statement should either include the hypothesis or the degenerate case must be treated separately.","section":"Abstract and Theorem 4.10, Remark 4.13"},{"comment":"The 'more precise version' of Theorem 1.2 stated as Theorem 3.20 claims equivalence with condition (1) 'there is a positive q such that P_ℓ is a q-Boltzmann map', while Theorem 1.2 allows 0 ≤ q ≤ q_c. The proof of (3)⇒(2) shows that all trees have equal probability, but the subsequent (2)⇒(1) argument divides by P_ℓ(Q=q) and defines q(ℓ) as a ratio of such probabilities; this is undefined when the measure is supported on trees. The tree-supported case either needs to be included in the statement as the q=0 Boltzmann law, or the theorem must exclude it. As it stands, condition (3) holds for uniform tree-supported measures while condition (1), with positive q, fails, so the equivalence is not correct as stated for the degenerate case.","section":"Theorem 3.20 and proof, §3.3"},{"comment":"The metric Markov property is stated using the relation eq ⊂ eQ, but the text explicitly says in §5.2 that {eq ⊂ eQ} has probability zero for Boltzmann metric maps, while {eq ≺ eQ} has positive probability. The proof of Theorem 5.11 indeed conditions on eq ≺ eQ and only at the end writes eq ⊂ eQ, and Definition 5.24 also uses eq ⊂ eQ. Since conditioning on a null event is not well-defined without further specification, the metric Markov property needs to be restated with the active-submap relation ≺, which is the relation that makes the weak Markov property a statement about positive-probability events. This is a load-bearing issue for the metric characterization Theorem 5.25.","section":"Theorem 5.11 and Definition 5.24, §5.2"},{"comment":"The base cases of the tree-comparison inductions are not fully proved. In Claim 4.12 the positivity of q4 is established by the sentence 'if q4 is 0 there have to be ...' and the justification is left to the reader; this is exactly the point where the non-degeneracy assumption is needed. In the metric case, Claim 5.31 says that the base case follows by the same argument as in Theorem 4.10 and leaves it to the reader, and the proof of Theorem 5.25 similarly defers the proof that q(ℓ,b) does not depend on ℓ to 'directly mimics the proof of Theorem 4.10'. The introduction states the paper is completely self-contained, but these deferred arguments are load-bearing for the decorated and metric characterizations and should be written out.","section":"Claim 4.12 and Claim 5.31, §4.3 and §5.5"}],"minor_comments":[{"comment":"The definition of stopping map includes Q ⊂ Q almost surely; the remark explaining why this is needed in the spatial setting is helpful, but it would be clearer to state explicitly that the index set for filtrations is countable in the discrete case.","section":"Section 3.2, Definition 3.9 and Remark 3.10"},{"comment":"In the (2)⇒(1) part, the notation q(1) is introduced only at the last line; it would be clearer to define q := q(1) before the final display.","section":"Section 3.3, proof of Theorem 3.20"},{"comment":"The definition of phantom exterior faces says 'the phantom faces F_i(q)' in one sentence, which appears to be a typo for F_e(q).","section":"Section 4.1"},{"comment":"The normalization constant W^{ℓ,b}_{λ,q,β} is written with the arguments in different orders in the text and display; please unify the notation for q, λ, and β.","section":"Section 5.3, Definition 5.4"},{"comment":"The proof of Theorem 5.11 is written for one-dimensional decorations, but the theorem statement says R_n in the surrounding discussion; please specify the dimension used in each statement.","section":"Section 5.4.1, Theorem 5.11"},{"comment":"The proof of Proposition 7.2 is diagram-based and refers to a case-by-case argument; a fully formalized description of the branching cases would make the counterexample easier to verify.","section":"Section 7, Proposition 7.2"}],"recommendation":"major_revision","confidential_remarks":"The paper contains substantial original material and the main proof architecture seems sound in the non-degenerate cases, but the advertised abstract statement overreaches and the metric Markov property is currently stated with a null conditioning event. The editorial decision should ask the authors to fix the abstract and theorem statements, include the degenerate tree-supported case or explicitly exclude it, and write out the deferred base-case arguments. The paper is likely salvageable within the scope of a major revision; I do not see grounds for rejection if those issues are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know about this one: it is a serious, mostly self-contained paper on Markov properties for quadrangulations, and the central claims are probably right. The undecorated characterization (Markov plus rerooting invariance implies Boltzmann) is proved cleanly, and the stopping-map/local-map equivalence together with the Section 7 counterexample are real additions. The decorated and metric analogues are new, but they carry assumptions that the abstract forgets to mention.\n\nWhat is genuinely good: the Boltzmann weight is derived from ratios of probabilities, not assumed, so there is no fitting or circularity. The proof architecture—first show uniformity on fixed face counts, then derive the constant ratio—is coherent. The local-map section transfers GFF ideas to maps in a natural way, and the example of a stopping map not obtainable by peeling is concrete and convincing.\n\nWhere the paper is soft: the abstract states the decorated characterization without caveat, and that statement is false as written. Theorem 4.10 requires P_{ell,b}(Q has at least one face) > 0; without it, tree-supported measures satisfy Markov and rerooting invariance but are not Boltzmann-decorated. The authors know this—Remark 4.13 concedes exactly that—but the abstract and introduction still overreach. This is not cosmetic: the proof that q4 > 0 in Claim 4.12 is exactly where the non-degeneracy is used, and the argument is compressed. The metric theorem similarly assumes continuity in boundary conditions and positive edge lengths with high probability; these are reasonable technical conditions, but they should be stated where the main theorem is advertised.\n\nI also flagged several steps left to the reader (parts of Claim 4.12, Claim 5.31, and the metric analogue in Theorem 5.25). I did not find a fatal error, and the main lines look right, but a referee will need to see those arguments written out before the characterizations can be considered fully verified.\n\nWho this is for: researchers working on random planar maps, peeling, and decorated or metric maps. It deserves a serious referee. With a revision that fixes the abstract, states the non-degeneracy assumptions up front, and fills in the deferred proofs, I would expect acceptance.\n\nRecommendation: send it to review. The conditional verdict is appropriate; the theorems as stated in the body are likely correct, and the gap is presentation and completeness rather than a load-bearing flaw.","headline":"Serious, mostly sound paper whose abstract overclaims the decorated characterization; the body's theorems are likely correct and deserve refereeing.","tokens_in":48015,"tokens_out":2697,"would_cite":true,"duration_ms":29554,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60D05","05C80","60G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the Markov property together with rerooting invariance forces a planar map law to be a Boltzmann map, with probability proportional to $q^{|F|}$ for internal faces, and extends the same uniqueness to spin-decorated…","keywords":["planar maps","Markov property","Boltzmann maps","peeling","stopping maps","spin-decorated maps","metric maps","Brownian bridges"],"falsifier":"Take any rerooting-invariant family that satisfies the Markov property and compute the ratio $P(q)/P(\\bar q)$, where $\\bar q$ is $q$ with one extra face glued at the root. The theorem predicts this ratio is a constant $q$ independent of $q$ and of the boundary length; finding any family where the ratio depends on the shape of $q$ disproves the characterization. For the decorated theorem, the paper's own tree-supported example is a clean falsifier of the unqualified statement: a measure supported on trees with arbitrary rerooting-invariant weights satisfies the Markov property vacuously and is not a spin-decorated Boltzmann map.","tokens_in":47020,"feed_emoji":"🗺️","tokens_out":8924,"duration_ms":90846,"temperature":0.7,"pith_summary":"The paper sets out to prove that on quadrangulations with boundary, the spatial Markov property plus invariance under rerooting is strong enough to force the law to be a Boltzmann map: probability proportional to $q^{|F|}$, the number of internal faces, for a single parameter $0 \\le q \\le 1/12$. It establishes this for undecorated quadrangulations, for spin-decorated maps such as Ising or Gaussian-free-field decorations on faces, and for decorated metric maps where edges are intervals with random lengths and the decoration runs along them. It also proves a converse: every random submap that induces a Markovian decomposition is a stopping map for some filtration indexed by submaps, and it exhibits a stopping map that no one-face-at-a-time peeling algorithm can produce. The interest is that the Markov property is the main tool for exploring random maps, and the paper shows which laws and which stopping sets are actually compatible with it.","feed_headline":"Markov property forces Boltzmann weights on planar maps","feed_subtitle":"The same twin assumptions force Boltzmann weights on spin-decorated and metric quadrangulations too.","key_machinery":"The load-bearing object is the gluing relation $q \\subset Q$ between a quadrangulation with holes and the full map, which makes the space of maps with holes an index set for filtrations. A random submap $\\mathcal{Q}$ is a stopping map when $\\{\\mathcal{Q} \\subset q\\}$ is measurable with respect to the $\\sigma$-algebra of $q$; the strong Markov property says that conditionally on $\\mathcal{Q}$, the fillings of its holes are independent Boltzmann maps. The proof of uniqueness works by comparing a map $q$ with the map $\\bar q$ obtained by adding one face at the root, forming the ratio $P(q)/P(\\bar q)$; writing that ratio through two different peeling routes shows it is a constant $q$, and then the Markov property upgrades this to $q^{|F(q)|}$. In the metric case the same ratio argument runs through densities with respect to Lebesgue measure, using the identity $\\int \\hat{P}^{u,z}_{w_1} \\hat{P}^{z,v}_{w_2} dz = \\hat{P}^{u,v}_{w_1+w_2}$ for non-normalised Brownian bridges, which transfers the Markov property across a cut made mid-edge.","core_discovery":"The central discovery is a uniqueness theorem: if $(P_\\ell)_\\ell$ is a sequence of laws on quadrangulations with boundary satisfying the Markov property (conditioning on a submap leaves independent Boltzmann-distributed fillings in each hole) and rerooting invariance, then for every map $q$, $P_\\ell(q) \\propto q^{|F(q)|}$ with $q \\le q_c = 1/12$. The same conclusion holds for spin-decorated maps, where the density acquires the Gibbs factor $\\exp(-\\frac{\\beta}{2}\\sum_{i\\sim j}\\|\\sigma_i-\\sigma_j\\|^2)$, and for decorated metric maps, where the density is $q^{|V|} \\prod_e e^{-\\lambda w_e} \\hat{P}^{\\sigma_i\\sigma_j}_{w_e} dw_e$ with $\\hat{P}$ the non-normalised Brownian bridge measure; exploring mid-edge produces the $w_e^{-n/2}$ factor from a Brownian bridge returning to its starting value. The paper further proves that a random submap induces a Markovian decomposition of a Boltzmann map if and only if it is a stopping map for a suitable filtration, and gives an explicit stopping map that cannot be recovered by peeling one face at a time. For decorated maps the theorem assumes at least one internal face has positive probability, without which the characterization can fail on tree-supported measures; the metric version assumes positive edge lengths with high probability and continuity in boundary conditions.","pith_inferences":["Because the uniqueness argument only needs two peeling routes and a density ratio, a testable extension is to run the same ratio on maps with several boundary components or on bipartite maps with face weights: the predicted outcome is again a one-parameter Boltzmann law, with an extra weight per boundary component if rerooting is relaxed.","The $w^{-n/2}$ factor is the paper's explanation of why small edges are cheap: it is the Brownian bridge density of returning to the starting spin. One can test this directly by simulating a Boltzmann decorated metric map and measuring the empirical edge-length density against $w^{-n/2} e^{-\\lambda w}$; deviations would show where the Brownian-bridge model stops being the right decoration.","In the degenerate tree-supported case, the Markov property becomes vacuous because there are no interior faces to condition on; this suggests that for decorated models the natural state space should exclude maps with no interior faces if one wants a clean uniqueness theorem.","The local-maps-are-stopping-maps direction transfers naturally to continuum random geometries: cutting a Liouville quantum gravity surface by a stopping set should be Markovian exactly for sets that are local in the filtration sense, mirroring the discrete theorem."],"forward_implications":["Any rerooting-invariant, Markovian law on quadrangulations with boundary is Boltzmann with a single parameter $q \\in [0, 1/12]$; no other parameters survive once the two symmetries are imposed.","The strong Markov property is available for every stopping map of a Boltzmann map, so exploration is not restricted to the classic one-face peeling procedure.","Conversely, the only random submaps that give a Markovian decomposition are stopping maps, so the two notions are exactly equivalent.","There exist stopping maps that no algorithmic one-face peeling can generate, and such maps still yield Markovian decompositions.","Boltzmann decorated metric maps are the unique Markov rerooting-invariant laws on metric maps, with an explicit edge-length density containing $e^{-\\lambda w}$ times the $w^{-n/2}$ Brownian-bridge return factor."],"supporting_citations":[{"why":"Defines Boltzmann maps and their peeling Markov decomposition, the framework both characterization theorems refine.","marker":"[Cur19]"},{"why":"Gives the weakly Markovian characterization the paper contrasts with, since its hypotheses differ from rerooting-invariant Markov laws.","marker":"[BL21]"},{"why":"Provides the conditional-density lemma used to decompose decorated and metric maps along holes.","marker":"[Kal01]"},{"why":"Introduces local sets of the Gaussian free field, the model for the local-map couplings shown to be stopping maps.","marker":"[SS13]"},{"why":"Supplies the inspiration for the map which is a stopping map but not obtainable by one-face peelings.","marker":"[MS16]"},{"why":"Defines generalised Ising systems, the spin-decorated measure that decorated Boltzmann maps are built from.","marker":"[New74]"},{"why":"Introduces the metric-graph Gaussian free field Markov property that motivates Boltzmann decorated metric maps with Brownian bridges on edges.","marker":"[Lup16]"}],"fun_headline_variants":["Markov property forces Boltzmann-type map laws","Boltzmann maps are the only Markovian planar maps","Uniqueness: Markov plus rerooting gives Boltzmann weights","Markovian planar maps must be Boltzmann","Markov property pins down Boltzmann-type laws"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For decorated maps the characterization needs at least one internal face to have positive probability, because if the map is almost surely a tree the decoration is empty and non-Boltzmann rerooting-invariant Markov laws exist; the metric version moreover needs edge lengths positive with high probability and boundary continuity.","fun_headline_variants_meta":{"raw":{"variants":["Markov property forces Boltzmann-type map laws","Boltzmann maps are the only Markovian planar maps","Uniqueness: Markov plus rerooting gives Boltzmann weights","Markovian planar maps must be Boltzmann","Markov property pins down Boltzmann-type laws"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000631,"raw_usage":{"total_tokens":2924,"prompt_tokens":964,"completion_tokens":1960,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":1889}},"tokens_in":580,"tokens_out":1960,"duration_ms":13852,"temperature":1.0,"reasoning_tokens":1889,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:04:23.179680+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any rerooting-invariant family that satisfies the Markov property and compute the ratio $P(q)/P(\\bar q)$, where $\\bar q$ is $q$ with one extra face glued at the root. The theorem predicts this ratio is a constant $q$ independent of $q$ and of the boundary length; finding any family where the ratio depends on the shape of $q$ disproves the characterization. For the decorated theorem, the paper's own tree-supported example is a clean falsifier of the unqualified statement: a measure supported on trees with arbitrary rerooting-invariant weights satisfies the Markov property vacuously and is not a spin-decorated Boltzmann map.","supporting_citations":[],"review_version":1}