{"id":"e2900719-522c-4185-972a-55653c84f3b7","arxiv_id":"2505.21823","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Globally centered discrete snakes on size-conditioned critical Bienaymé trees converge, after rescaling, to the Brownian snake, with heavy displacement tails producing hairy tour limits.","lead":"This paper proves scaling limits for discrete snakes, which record random spatial positions along the branches of random family trees, under a global centering condition. It also settles the exact tail conditions needed for uniform convergence to the Brownian snake and gives new limit theorems for heights of trees and looptrees.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 1.1 is internally coherent; the third-moment assumption is explicit, not claimed necessary, and the proof steps relying on it check out.","rationale":"The reader's weakest assumption is the same spot I scrutinized: E[xi^3]<\\infty in [A2]. But a superfluous sufficient condition is not an error. I traced all three appearances: Lemma 6.3, Lemma 6.2, and Proposition 5.10. The asymptotics in Lemma 6.3 are correct; the tail bound uses E[xi^3]; and Lemma 6.2's bound (6.2) rests on a uniform comparison that is standard but not fully written out. Since this is a proof-completeness issue rather than a false central claim, the ACCEPT verdict should stand unchanged.","tokens_in":81428,"tokens_out":40546,"duration_ms":457441,"concrete_test":"Re-derive Lemmas 6.2 and 6.3 with an explicit proof of sup_{m,k\\le m} P(bD^m_1=k)/P(\\bar{\\xi}=k)<\\infty under E[xi^3]<\\infty, using Kemperman's formula and Petrov's local CLT. If this ratio is unbounded, replace the comparison in (6.5) with a direct bound; otherwise the tightness induction in Proposition 5.10 stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the central argument as: random finite-dimensional convergence under [A1] via the line-breaking construction, then uniform convergence under [A2] via the Haas–Miermont tightness scheme. The only place I would push is the third moment condition E[xi^3]<\\infty in [A2]. The authors themselves flag it as possibly an artefact, and the reader identifies it as the weakest assumption. It is not, however, a correctness threat: the theorem states it as a sufficient hypothesis, and no part of the theorem claims it is necessary. Checking the specific uses: Lemma 6.3's total-variation bound d_TV(bD^m_1, \\bar{\\xi})=o(m^{-1/2}) is obtained correctly from Petrov's expansion; the O(1/m) contribution from k\\le m^{1/4} is indeed o(m^{-1/2}), and the tail is controlled by E[xi^3]<\\infty. The one assertion that is used without proof is the uniform comparison P(bD^m_1=k)\\le cP(\\bar{\\xi}=k) for k\\in[m] (Lemma 6.3 proof and Lemma 6.2). This is a standard consequence of the local CLT and large-deviation bounds for aperiodic finite-variance integer random walks, so I do not regard it as a fatal gap, but it is the most worthwhile point to verify formally. Apart from that, the line-breaking construction, the random-finite-dimensional convergence, and the tightness decomposition into typical/mid-range/large displacements are internally consistent.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves scaling limits for the head of a discrete snake built from a branching random walk whose genealogy is a critical Bienaymé tree conditioned to have n vertices, under a 'global centering' condition on the displacement of a size-biased uniform child. Theorem 1.1 states that under finite global variance β^2 the rescaled height and spatial location processes converge, in the sense of random finite-dimensional distributions, to (2/σ e, β√(2/σ) r), where e is a Brownian excursion and r is the head of the Brownian snake driven by e; uniform functional convergence is obtained under the additional conditions E[ξ^3]<∞ and P{max_{i≤ξ}|Y_{ξ,i}|>y}=o(y^{-4}). Proposition 1.2 shows the tail condition is necessary. Theorems 1.4 and 1.5 give hairy-tour limits when the displacements have heavier tails, and Corollaries 1.8 and 1.9 apply the main result to the difference between the height process and the Łukasiewicz path and between the height process and the looptree height process. The proofs use a discrete line-breaking construction, a change of measure for the size-biased degree sequence, and a Haas–Miermont-style tightness scheme.","tokens_in":81764,"tokens_out":5383,"duration_ms":56281,"significance":"If correct, this is a substantial advance: it removes the bounded-support hypothesis in Marckert's earlier global-centering result, extends the Janson–Marckert discrete-snake convergence to sibling-dependent displacements under weak moment assumptions, and identifies the exact tail condition needed for uniform convergence. The line-breaking convergence in Proposition 4.1 and Corollary 4.2 is a clean and reusable tool, and the authors are appropriately explicit about the role of the finite third moment, stating openly that they do not know whether it is necessary. The two applications to height-process differences are natural and the necessary-and-sufficient tail statements in Corollaries 1.8 and 1.9 are appealing. The paper also contains explicit, falsifiable limit statements for the heavy-tailed cases, which should be of interest beyond the immediate branching-random-walk community.","major_comments":[{"comment":"The comparison P{bD_m^1=k} ≤ cP{ξbar=k} for k∈[m] is asserted without proof and is then used to control the tail of the total variation sum and, through Lemmas 6.2 and 6.4, to prove the key tail bound Proposition 5.10. This is not immediate from Kemperman's formula because the ratio P{S_{m-1}=m-1-k}/P{S_m=m-1} must be bounded uniformly in k. Please provide a proof or a precise reference; as written, a load-bearing step of the tightness argument is unverified.","section":"Section 6, proof of Lemma 6.3"},{"comment":"The proof says that the proof of Proposition 5.10 'generalises immediately' after replacing n^{1/4} by n^{1/(4-η)}. Proposition 5.10 is the central tail estimate, and the hypotheses in Section 7 are [A3] rather than [A2], so the induction in Section 6 does not formally apply without checking the analogues of Lemmas 6.1–6.4. Since Theorems 1.4 and 1.5 depend on this step, the generalized lemma and its proof, or at least a precise statement with the required modifications, should be included.","section":"Section 7.1, Proposition 7.3"},{"comment":"The identity 'conditionally on J_k, d|γ^k| = B^2/(J_k^2+B^2)' is used to obtain the k→∞ limit in (5.11). Here d|γ^k| was defined as a size-biased pick from the list (|γ^k_j|, j≥1), but the displayed formula is a normalized quantity in [0,1]. Please clarify the normalization of the γ^k_j's and state precisely which result from [36] is being used; as it stands, the passage from the unnormalized size-biased pick to the expression B^2/(J_k^2+B^2) is not fully justified.","section":"Section 5.2, proof of Proposition 5.7"}],"minor_comments":[{"comment":"There is a typo: 'convergence in the first coordinate in in C([0,1], R^2)' should read 'convergence in the first coordinate in C([0,1], R^2)'.","section":"Section 7.2, Proposition 7.6"},{"comment":"The hypotheses are written as µ=(µ_k)_{k≥1}, whereas the offspring distribution µ is indexed from k≥0 elsewhere in the paper; please make the indexing consistent.","section":"Corollaries 1.8 and 1.9"},{"comment":"The quantitative local CLT is stated for k∈N with the comment that the generalization from [3, Lemma 5.5] is standard; it would be helpful to include a one-sentence indication of how the uniformity in k is obtained.","section":"Appendix A.1, Lemma A.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically strong and likely correct in its main claims. The main reason for major revision is not a doubt about the central result but the presence of two load-bearing steps that are currently asserted rather than proved: the uniform comparison in Lemma 6.3 and the claimed immediate generalization of Proposition 5.10 to the heavy-tailed setting. Both are probably routine for specialists, but they support the tightness arguments and should be made explicit. I would support acceptance after these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is the real thing. It completes a natural program in discrete snake convergence by removing Marckert's bounded-support restriction and replacing his p > 4 moment condition with finite global variance, plus a finite third moment when uniform convergence is wanted. It also proves the tail condition o(y^-4) is necessary, which was previously known only as sufficient, and it extends the hairy-tour limits to dependent displacement vectors. The two applications, the height/Lukasiewicz difference and the looptree height difference, are substantive corollaries and not decoration.\n\nThe proof is organized the way a good modern probability paper should be: random finite-dimensional convergence goes through a discrete line-breaking construction that converges to Aldous's continuous one, and tightness adapts the Haas-Miermont scheme for Markov branching trees. The necessity proposition is clean. I read the central argument as coherent and internally consistent. The removal of bounded support is genuine: the paper handles unbounded offspring distributions, where Marckert's method stalls.\n\nThe soft spots are minor and mostly ones the authors already acknowledge. The third moment E[xi^3]y\\}\\sim c y^{-4}$, so the tail condition fails. Corollary 1.8 predicts that $n^{-1/4}\\max_{0\\le i\\le n}|\\sigma H_n(i)-2\\sigma^{-1}W_n(i)|$ does not converge to the maximum modulus of $\\beta\\sqrt{2/\\sigma}\\,r$; a direct simulation or exact computation of that maximum should show macroscopic $\\Theta(n^{1/4})$ fluctuations with probability bounded away from zero, confirming the necessity claim of Proposition 1.2.","tokens_in":81254,"feed_emoji":"🐍","tokens_out":9409,"duration_ms":88700,"temperature":0.7,"pith_summary":"The paper establishes a scaling limit for discrete snakes built on size-conditioned critical Bienaymé trees, when the spatial displacements are globally centered rather than locally centered. Under a finite global variance condition, the rescaled head of the discrete snake converges, in random finite-dimensional distributions, to the head of the Brownian snake driven by a Brownian excursion. Adding a finite third moment and a tail condition upgrades this to uniform functional convergence. The paper also proves the tail condition is necessary, and describes heavy-tailed alternatives in which the limit is a 'hairy tour' decorated by intervals or pure jumps. Two applications give scaling limits for the difference between the height process and the Łukasiewicz path, and for the difference between a tree's height process and its looptree's height process.","feed_headline":"One centering rule sends discrete snakes to Brownian snake","feed_subtitle":"For critical conditioned trees, global centering gives a Brownian snake limit; heavy tails give the hairy tour.","key_machinery":"A bijection between permutations of edge labels and labeled ordered rooted trees, which is a discrete analogue of the continuous line-breaking construction of the Brownian tree, carries the argument. Under this bijection, the subtree spanned by the root and $k$ uniform vertices is built from $k$ paths whose lengths and attachment points converge, after rescaling by $\\sqrt{n}$, to the jump times and attachment points of the continuous line-breaking construction. Along each path, partial sums of the displacements behave like a random walk with IID steps distributed as $Y_{\\bar{\\xi},U}$, so they scale to Brownian motion with diffusivity $\\beta$ after $n^{1/4}$ rescaling; displacements at branch points are shown to be $O_P(1)$ and hence negligible on that scale. Tightness is obtained by truncating displacements into typical, mid-range and large parts, and bounding the maximum spatial location through an induction that relies on a total-variation estimate between the degree of the root of the conditioned tree and the size-biased offspring law.","core_discovery":"The central claim is that the spatial locations encountered during a depth-first exploration of a size-conditioned critical Bayes tree have a universal Brownian scaling limit once displacements are centered in the global sense, meaning $E[Y_{\\bar{\\xi},U}]=0$ with finite global variance $\\beta^2<\\infty$. Concretely, Theorem 1.1 states that under this condition the joint rescaled height, head, contour and contour-head processes converge in random finite-dimensional distributions to $$\\left(\\frac{2}{\\$\\sigma$}e_t,\\ \\$\\beta$\\sqrt{\\frac{2}{\\$\\sigma$}}r_t,\\ \\frac{2}{\\$\\sigma$}e_t,\\ \\$\\beta$\\sqrt{\\frac{2}{\\$\\sigma$}}r_t\\right)_{0\\le t\\le1},$$ where $e$ is a normalized Brownian excursion and $r$ is the head of the Brownian snake driven by $e$ (with covariance $\\min_{u\\in[s\\wedge t,s\\vee t]} e_u$). Under the additional assumption $E[\\xi^3]<\\infty$ and the tail condition $P\\{\\max_{1\\le i\\le\\xi}|Y_{\\xi,i}|>y\\}=o(y^{-4})$, the convergence is uniform in $C([0,1],\\mathbb{R}^4)$. The tail condition is sharp: Proposition 1.2 shows that if $\\limsup_{y\\to\\infty} y^4P\\{\\max_{1\\le i\\le\\xi}|Y_{\\xi,i}|>y\\}>0$, then displacements of order $n^{1/4}$ persist in the tree, so no continuous limit exists. Under a further regular-variation assumption [A3], the large displacements organize into a Poisson-decorated 'hairy tour' when $\\eta=0$, or a pure-jump decoration when $\\eta\\in(0,2)$.","pith_inferences":["A natural stress test is whether uniform convergence survives when $E[\\xi^3]=\\infty$ but the tail condition still holds; the authors' own unresolved comment identifies this as the most direct route to deciding if the third moment is an artefact of the tightness proof.","The global-centering identity $E[Y_{\\bar{\\xi},U}]=0$ can be read as a recipe: centering in the size-biased coordinate is exactly what makes branch-point displacements vanish on the $n^{1/4}$ scale, so other centering schemes should be compared against this coordinate.","The hairy-tour theorems suggest that point-process decorations of the Brownian snake are the general heavy-tailed limit, and the deterministic family $Y_{k,j}=\\sigma-(2/\\sigma)(k-j)$ gives an explicit test case where the decoration is a full interval rather than isolated points.","The looptree application indicates that differences of two height-like encodings are generically globally centered snakes; similar differences for other graph encodings, such as dual trees or spanning trees, may admit the same theorem with explicit $\\beta^2$ computations."],"forward_implications":["Under [A1], the rescaled height process and the head of the discrete snake converge jointly in random finite-dimensional distributions to $(2\\sigma^{-1}e,\\ \\beta\\sqrt{2/\\sigma}\\,r)$; adding [A2] upgrades this to uniform convergence in $C([0,1],\\mathbb{R}^4)$.","The tail condition $P\\{\\max_{1\\le i\\le\\xi}|Y_{\\xi,i}|>y\\}=o(y^{-4})$ is necessary: when it fails, displacements of order $n^{1/4}$ persist at vertices with positive asymptotic probability, ruling out a continuous limit.","Under [A3] with $\\eta=0$, the limit becomes the hairy tour: the continuous curve $\\beta\\sqrt{2/\\sigma}\\,r$ decorated by vertical intervals whose endpoints form a Poisson process with intensity $dt\\otimes\\pi$; with $\\eta\\in(0,2)$ the limit is the pure-jump decoration $U(0,\\Xi)$.","The difference $\\sigma H_n-2\\sigma^{-1}W_n$ between the height process and the Łukasiewicz path converges in $C([0,1],\\mathbb{R}^2)$ if and only if $P\\{\\xi>y\\}=o(y^{-4})$, with explicit global variance $\\beta^2=\\frac{4}{3\\sigma^2}(E[\\xi^3]-1)-(\\sigma^2+2)$.","The difference $cH_n-H_n^\\circ$ between a tree's height process and its looptree's height process converges uniformly, with the same necessary and sufficient tail condition."],"supporting_citations":[{"why":"Supplies the continuous line-breaking construction of the Brownian tree, which is the limit object the discrete bijection converges to.","marker":"[4]"},{"why":"Establishes the equivalence of height and contour parametrisations used to reduce the joint convergence to the head process.","marker":"[12]"},{"why":"Provides the Markov branching tree tightness framework and the split-probability estimate used in the induction for the maximum spatial location.","marker":"[14]"},{"why":"Prior discrete snake convergence with IID displacements, and the origin of the hairy tour/jumping snake limit extended here.","marker":"[18]"},{"why":"Proves the globally centered snake limit under bounded offspring support, the result this paper generalizes by removing boundedness.","marker":"[29]"},{"why":"Homeomorphism theorem connecting the head of the snake to the full snake path process.","marker":"[31]"},{"why":"Convergence of the height, contour and Łukasiewicz processes; the height–Łukasiewicz difference application strengthens it.","marker":"[32]"},{"why":"Local central limit theorem used to control the total variation distance between the conditioned root degree and the size-biased offspring law.","marker":"[35]"}],"fun_headline_variants":["Global centering sends discrete snakes to Brownian limits","Centered snakes converge to Brownian, heavy tails to hairy","Snake scaling limit: Brownian under centering, hairy with heavy tails","Global centering yields Brownian snake, heavy tails yield hairy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem needs the offspring distribution to have a finite third moment to make the tightness argument work, and the authors state they do not know whether this condition is genuinely necessary or only an artefact of their proof.","fun_headline_variants_meta":{"raw":{"variants":["Global centering sends discrete snakes to Brownian limits","Centered snakes converge to Brownian, heavy tails to hairy","Snake scaling limit: Brownian under centering, hairy with heavy tails","Global centering yields Brownian snake, heavy tails yield hairy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000282,"raw_usage":{"total_tokens":1748,"prompt_tokens":1103,"completion_tokens":645,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":719,"completion_tokens_details":{"reasoning_tokens":573}},"tokens_in":719,"tokens_out":645,"duration_ms":6604,"temperature":1.0,"reasoning_tokens":573,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:23:10.977810+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the deterministic displacement family $Y_{k,j}=\\sigma-(2/\\sigma)(k-j)$ with a critical offspring distribution whose tail satisfies $P\\{\\xi>y\\}\\sim c y^{-4}$, so the tail condition fails. Corollary 1.8 predicts that $n^{-1/4}\\max_{0\\le i\\le n}|\\sigma H_n(i)-2\\sigma^{-1}W_n(i)|$ does not converge to the maximum modulus of $\\beta\\sqrt{2/\\sigma}\\,r$; a direct simulation or exact computation of that maximum should show macroscopic $\\Theta(n^{1/4})$ fluctuations with probability bounded away from zero, confirming the necessity claim of Proposition 1.2.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the continuous line-breaking construction of the Brownian tree, which is the limit object the discrete bijection converges to."},{"cited_title":"and LE GALL , J.-F","cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence of height and contour parametrisations used to reduce the joint convergence to the head process."},{"cited_title":"and M IERMONT , G","cited_arxiv_id":null,"evidence_quote":"Provides the Markov branching tree tightness framework and the split-probability estimate used in the induction for the maximum spatial location."},{"cited_title":"and MARCKERT , J.-F","cited_arxiv_id":null,"evidence_quote":"Prior discrete snake convergence with IID displacements, and the origin of the hairy tour/jumping snake limit extended here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the globally centered snake limit under bounded offspring support, the result this paper generalizes by removing boundedness."},{"cited_title":"and M OKKADEM , A","cited_arxiv_id":null,"evidence_quote":"Homeomorphism theorem connecting the head of the snake to the full snake path process."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Local central limit theorem used to control the total variation distance between the conditioned root degree and the size-biased offspring law."}],"review_version":1}