{"id":"efc99fd3-eddb-4a33-a165-0ea78f25ae3f","arxiv_id":"2412.15895","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For percolation on T times Z^d and on lamplighter graphs over a tree, the Hausdorff dimension of cluster limit sets jumps from at most 1/2 to 1 at the uniqueness threshold, and four critical thresholds coincide.","lead":"This paper studies random connection models on graphs built from a regular tree multiplied by a lattice, and proves a sharp jump at the uniqueness threshold: the boundary dimension of infinite clusters jumps from at most 1/2 to 1. It also shows that several critical thresholds, including the L2 boundedness threshold, coincide with the uniqueness threshold for these graphs, the first nonamenable examples with this property.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exact dimension formula in Theorem 1.1 depends on the unproved lower-bound half of Lemma 2.4 ('we omit further details here'); without it only the upper bound δ_H≤1/2 below p_u remains.","rationale":"The reader's weakest_assumption is exactly the point I would stress. I read Lemma 2.4 as the hinge of the paper: all of Theorems 1.1 and 1.4 pass through the identification of Hausdorff dimension with 1-β*_p. The upper bound is clean. The lower bound is deferred, and the reference to Lemma 2.8 is not a substitute, since Lemma 2.8's embedded branching process is used for a different purpose (control of Q_p(n) and backscattering) and does not by itself yield a boundary measure. A secondary concern is the reliance on [26] for Lemmas 2.1 and 2.6, but that is an external-theorem issue: if the companion paper is sound, this is not a mathematical gap in the present text. The lower-bound omission, by contrast, is internal to the paper. I do not think this warrants rejection: the construction is plausibly fillable, and the upper-bound/jump result is already a significant theorem. But as written, the exact dimension formula is conditional, so I keep the reader's CONDITIONAL verdict.","tokens_in":17583,"tokens_out":18716,"duration_ms":167809,"concrete_test":"Write the omitted lower-bound proof for G=T×Z. Specifically, following [18,33], define embedded Galton-Watson processes whose individuals are open connections from o to fibers at levels n_j, use Lemma 2.2/2.8 to show the expected population at level n is (k-1)^{(1-β*_p)n+o(n)}, and apply the Hawkes-Lyons theorem to the induced subtree of occupied cylinders to get dim_H(Λ)≥1-β*_p. Verify also that this applies to every infinite cluster via the Lyons-Schramm indistinguishability step used in Lemma 2.4. If the construction needs an extra hypothesis (e.g. p>p_c or p≤p_u) that is not already stated, the theorem must be weakened accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the exact equality δ_H(Λ)=1-β*_p on (p_c,p_u], not merely a dimension jump. Lemma 2.4 is the only place where Hausdorff dimension is connected to β*_p. The upper bound is proved via the cover C_n and Markov's inequality. The lower bound is not proved: the text says a 'very similar' argument to [18,33] is used and 'we omit further details here,' deferring to Lemma 2.8 for a 'similar construction.' That deferral does not close the gap. Lemma 2.8 constructs an embedded branching process that controls Q_p(n), the probability of a single connection from o to a fiber at level -n; it does not construct the boundary measure or Frostman-type lower bound needed to conclude dim_H(Λ)≥1-β*_p. The upper bound alone gives δ_H≤1-β*_p≤1/2 for p<p_u (using Lemmas 2.7/2.9), so the abrupt jump to dimension 1 at p_u would survive, but the quantitative content of Theorem 1.1—the continuous increase from 0 to δ_H(p_u) and the identification δ_H=1-β*—would not be established. This is a fillable gap rather than a contradiction, but it is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Bernoulli percolation on the product T×H of a k-regular tree T with an infinite amenable Cayley graph H, and on the lamplighter graph LL(T). It claims that on the event of an infinite cluster, the Hausdorff dimension of the cluster's limit set in the tree boundary is at most 1/2 below the uniqueness threshold p_u and jumps to 1 above p_u. It also claims equality of several critical thresholds: the uniqueness threshold p_u, the L2-boundedness threshold p_{2→2}, the tiltability threshold p_t, and the heaviness threshold p_h. The proofs use nonunimodular percolation techniques, a supermultiplicative sequence (P_p(n)) with rate β*_p, and a Hammersley-Welsh argument to show χ_{p,1/2}<∞ for p<p_u.","tokens_in":17871,"tokens_out":16933,"duration_ms":141446,"significance":"If the claims hold, this constitutes the first proof of a dimension jump for unoriented percolation and the first nonamenable examples with p_{2→2}=p_u. The paper is well-structured and contains detailed arguments in Sections 2.2-2.4, including the backscattering lemmas and the Hammersley-Welsh decomposition. However, the central dimension formula rests on a lower bound in Lemma 2.4 that is explicitly omitted, and two key lemmas are imported from the same-authors' companion preprint [26]. The lower-bound gap is fillable in principle, and the companion-paper dependence is a matter of verification, so the work has substantial potential but is not yet complete as a standalone paper.","major_comments":[{"comment":"The lower bound δ_H(Λ) ≥ 1−β*_p for p_c<p≤p_u is asserted but not proved. The text states that the matching lower bound follows by a 'very similar' argument to [18,33] and 'we omit further details here,' deferring to Lemma 2.8. However, Lemma 2.8 constructs embedded branching processes that control the probability Q_p(n) of a single first-arrival connection to a fiber; it does not construct a boundary measure or a Frostman-type lower bound for the limit set. Without this lower bound, Theorem 1.1's exact formula δ_H(p)=1−β*_p, and the claimed continuous increase from 0 to δ_H(p_u), are not established. Only the upper bound δ_H≤1−β*_p≤1/2 below p_u would follow. This is a load-bearing gap in the proof of the main theorem.","section":"Section 2.1, Lemma 2.4"},{"comment":"Lemmas 2.1 and 2.6 are stated without proof and are attributed to the companion preprint [26]. Lemma 2.1 is used to identify p_u with the threshold for finite fiber intersections, and Lemma 2.6 provides the exponential tail estimate that is essential in Lemma 2.5 and in the proofs of Lemmas 2.7 and 2.9. Since [26] is a same-author preprint that is not yet a published reference, the present paper's central claims cannot be verified independently of [26]. The authors should either provide full proofs of these statements in an appendix or ensure that [26] is accepted and publicly available in a verifiable form before the paper is accepted.","section":"Sections 2.1–2.2, Lemmas 2.1 and 2.6"},{"comment":"In the proof of Lemma 2.9, the final step asserts that if the branching process Y^{h,g} survives, then 'in either case, the two clusters will become a.s. equal when we increase p to any p'>p in the standard monotone coupling.' This step is not proved. To justify it, one must argue that the infinitely many levels at which the clusters come within distance d(h,g) give rise to infinitely many edge-disjoint opportunities to connect the clusters at p', so that by Borel-Cantelli the merge occurs almost surely. This is plausible and likely fixable, but as written the merging argument is a genuine gap in the proof that β*_p≥1/2 for T×H.","section":"Section 2.3, Lemma 2.9"}],"minor_comments":[{"comment":"The statement contains the typo 'for each p_c < p c≤ p_u'; it should be 'for each p_c < p ≤ p_u'.","section":"Section 2.1, Lemma 2.4 (statement)"},{"comment":"The expression 'log e/Pp(n)' is ambiguous. It should be written as log(e/Pp(n)) to mean the natural logarithm of the ratio e divided by Pp(n).","section":"Section 2.2, Lemma 2.5"},{"comment":"There is a notation inconsistency: the sum is written over s∈S_{n,k}, while the set was previously defined as S_{l,n}. The index set should be S_{l,n} (or S_{n,l}), and the quantities should be labeled consistently.","section":"Section 2.4, proof of Lemma 2.10"},{"comment":"The lower bound 'by the Harris-FKG inequality' is not precisely a Harris-FKG statement; the argument is a union bound conditional on the event that at least one vertex in the fiber is connected, combined with independence of the extension edges. The sentence could be rephrased to avoid implying an FKG correlation inequality where none is needed.","section":"Section 2.1, proof of Lemma 2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically interesting and likely correct in its main ideas, but the missing lower bound in Lemma 2.4 is a significant gap in the proof of the central dimension theorem. In addition, the heavy reliance on the companion preprint [26] for Lemmas 2.1 and 2.6 means the manuscript is not self-contained. I would suggest asking the authors to either include the missing lower-bound proof or clearly indicate that it will appear in a separate paper, and to ensure that the companion paper is available for review. The backscattering and Hammersley-Welsh parts of the paper are quite detailed and give me confidence that the remaining gaps are fillable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main results are new and important: the dimension jump at p_u for unoriented percolation on T×H and LL(T), and the first nonamenable examples with p_{2→2}=p_u. The proofs use the nonunimodular framework plus a Hammersley–Welsh argument that are natural and mostly well executed. I agree with the reader: the architecture is credible and the central claims are very likely true, but the paper cannot be accepted as is.\n\nThe load-bearing soft spot is Lemma 2.4. The upper bound on the Hausdorff dimension of the limit set is proved, but the matching lower bound is not. The sentence \"we omit further details here\" and the pointer to Lemma 2.8 do not supply it: Lemma 2.8 constructs embedded branching processes that control the connection probability Q_p(n), not the boundary measure or Frostman-type estimate needed to show dim(Λ) ≥ 1−β*_p. Without that lower bound, Theorem 1.1's exact identity δ_H(p)=1−β*_p, the continuity on (p_c,p_u], and the value δ_H(p_u)≤1/2 are not established. The dimension jump from ≤1/2 to 1 would still follow from the upper bound alone, but the quantitative content would be lost. This is a fillable gap, but it is load-bearing.\n\nThe other issue is self-citation: Lemmas 2.1 and 2.6 come from the companion paper [26], and the full proof also relies heavily on [23]. That is not illegitimate, but it makes the paper not self-contained, and the referee should verify those dependencies.\n\nMinor points: Lemma 2.9's uniform lower bound on the survival probability of the embedded branching process is asserted a bit quickly, and Lemma 2.4 has a typo in its statement (\"p_c < p c ≤ p_u\"). Neither affects the math.\n\nWho this is for: percolation theorists, especially those working on nonamenable graphs and the branching random walk analogy. It deserves serious referee time, but the referee should insist that the lower bound in Lemma 2.4 be written out in full, or that the theorem be restated as a conditional result.","headline":"A genuinely new dimension jump for unoriented percolation on tree products, but the exact dimension formula rests on an omitted lower bound that must be supplied before the paper is complete.","tokens_in":18425,"tokens_out":2887,"would_cite":false,"duration_ms":25141,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B43","60J80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for percolation on tree-with-lattice graphs, the boundary dimension of an infinite cluster is at most 1/2 at the uniqueness threshold and jumps to 1 immediately above it.","keywords":["percolation","uniqueness threshold","Hausdorff dimension","nonunimodular transitive graphs","lamplighter graph","tree products","tilted susceptibility","branching random walk"],"falsifier":"Find a $p<p_u$ and an amenable $H$ for which the boundary limit set of an infinite cluster has Hausdorff dimension strictly larger than $1/2$; Theorem 1.1 says this cannot happen. A concrete route is to run percolation on $T\\times\\mathbb{Z}$ at a numerically sub-$p_u$ parameter, measure the box-counting or Hausdorff dimension of the cluster's accumulation set on $\\partial T$, and compare it with $1-\\beta_p^*$ estimated from the decay of $P_p(n)$; any systematic gap between the measured dimension and $1-\\beta_p^*$ below $p_u$ would falsify Lemma 2.4.","tokens_in":17348,"feed_emoji":"🌳","tokens_out":9187,"duration_ms":72818,"temperature":0.7,"pith_summary":"This paper studies Bernoulli percolation on the product of a regular tree with an infinite amenable graph, and on the lamplighter graph over the tree. It establishes that as the percolation parameter crosses the uniqueness threshold $p_u$, the set of accumulation points of an infinite cluster in the boundary of the tree changes dimension discontinuously: at most $1/2$ below and at $p_u$, and full dimension $1$ above $p_u$. The dimension is identified almost surely with $1-\\beta_p^*$, where $\\beta_p^*$ is an exponent extracted from slab-to-fiber connection probabilities. Along the way the paper proves that four central thresholds—$p_u$, the $L^2$ boundedness threshold, the tiltability threshold, and the heaviness threshold—coincide for these graphs, giving the first nonamenable examples known to have $p_{2\\to2}=p_u$. If correct, this makes the percolation transition on these graphs resemble the branching random walk recurrence transition in a quantitative, geometric way.","feed_headline":"Percolation clusters jump from half to full boundary dimension at p_u","feed_subtitle":"On tree×lattice graphs, infinite clusters leave boundary traces of dimension ≤1/2 at p_u; above it, dimension 1.","key_machinery":"The proof is carried by the exponent $\\beta_p^*$, defined from the supermultiplicative sequence $P_p(n)=P_p(o\\leftrightarrow[x])$ for a fiber at height $-n$, with $\\beta_p^*=-\\lim_n\\log P_p(n)/(n\\log(k-1))$; a standard supermultiplicativity argument gives the limit, and Lemma 2.4 ties it to geometry: the boundary limit set has Hausdorff dimension $1-\\beta_p^*$ on the event that the cluster is infinite. The nonunimodular automorphism group fixing an end of the tree supplies the tilted mass-transport principle, with modular function $\\Delta(x,y)=(k-1)^{h(x,y)}$, used to define tilted susceptibilities and the thresholds $p_t$ and $p_h$. Backscattering lemmas show $\\beta_p^*\\ge 1/2$ below $p_u$ by constructing paths that descend deep into the tree and return to the root fiber; for products with a lattice this uses an embedded branching random walk on the amenable factor and its spectral-radius behavior. Finally, a bridge-decomposition argument in the style used for self-avoiding walks bounds the tilted susceptibility $\\chi_{p,1/2}$ by a product of slab generating functions, proving $p_t=p_u$ and hence the threshold equalities.","core_discovery":"The central claim is Theorem 1.1: for percolation on $T\\times H$ with $H$ an infinite amenable Cayley graph, or on the lamplighter graph $LL(T)$, conditional on the cluster of the root being infinite, the Hausdorff dimension $\\delta_H(\\Lambda)$ of the set of limit points in $\\partial T$ of the projected cluster is $P_p$-almost surely equal to a constant $\\delta_H(p)$. This constant is $1-\\beta_p^*$ for $p_c<p\\le p_u$, with $\\delta_H(p_u)\\le 1/2$, and is $1$ for every $p>p_u$; moreover $p\\mapsto\\delta_H(p)$ is continuous on $(p_c,p_u]$. Theorem 1.4 adds $p_u=p_{2\\to2}=p_t=p_h$ for these graphs. The statement strengthens the known non-uniqueness at $p_u$: instead of merely many infinite clusters, each cluster's boundary footprint is geometrically small (dimension at most $1/2$) at and below the threshold and full (dimension $1$) above it.","pith_inferences":["Beyond the paper, the same mechanism should extend to products of trees with any amenable lamp group or with switch-walk-switch Cayley graphs, since the proof's key input is the amenability of fibers rather than the specific lattice structure.","The equality $p_{2\\to2}=p_u$ suggests a testable dichotomy: on graphs with $p_c<p_u$, the position of $p_{2\\to2}$ relative to $p_u$ may encode whether infinite clusters below $p_u$ are tree-like or dense, and simulations on hyperbolic tessellations versus $T\\times\\mathbb{Z}$ could calibrate this.","Because $\\beta_p^*$ is a supermultiplicative rate, one could estimate it from finite-slab simulations and use the predicted dimension jump as a numerical locator for $p_u$ on infinite-dimensional graphs.","The omitted lower-bound half of Lemma 2.4 is the part most worth checking: if the embedded branching-process construction cannot be made to give the matching dimension, the theorem would still hold as an upper bound but the exact dimension formula would be lost."],"forward_implications":["For every $p\\le p_u$, any infinite cluster's projected cluster accumulates on a boundary set of Hausdorff dimension at most $1/2$, so infinite clusters occupy at most a square-root fraction of the boundary.","For every $p>p_u$, the projected cluster is dense in the boundary and its accumulation set has full dimension $1$, matching the trivial upper bound.","The equality $p_{2\\to2}=p_u$ provides the first nonamenable transitive graphs for which the $L^2$ boundedness threshold sits at the uniqueness threshold rather than strictly below it.","The coincidence $p_u=p_t=p_h$ integrates the uniqueness threshold with the nonunimodular tiltability and heaviness thresholds for these graphs.","The dimension jump mirrors the branching random walk recurrence transition, giving a quantitative sense in which percolation on these graphs behaves like branching random walk up to $p_u$."],"supporting_citations":[{"why":"Supplies the exponential tail bound for fiber intersections and the characterization of $p_u$ via amenability of the fiber subgroup, used in Lemmas 2.1 and 2.6.","marker":"[26]"},{"why":"Provides the nonunimodular percolation framework, tilted susceptibilities, and the $p_c<p_t$ theorem that underpin the threshold equalities and continuity.","marker":"[23]"},{"why":"Establishes infinitely many infinite clusters at $p_u$ for $T\\times\\mathbb{Z}$, the result that Theorem 1.1 strengthens.","marker":"[43]"},{"why":"Supplies the branching-process lower-bound construction for Hausdorff dimension of limit sets invoked in Lemma 2.4.","marker":"[18, 33]"},{"why":"Develops the nonunimodular bridge-decomposition argument used to prove $\\chi_{p,1/2}<\\infty$ and hence $p_t=p_u$.","marker":"[20]"},{"why":"Gives non-uniqueness at $p_u$ for nonamenable products, used in Lemma 2.1's characterization.","marker":"[42]"}],"fun_headline_variants":["Boundary dimension of infinite clusters jumps from ≤1/2 to 1 at p_u","At p_u, cluster boundary traces jump from half to full dimension","Infinite cluster boundary: ≤1/2 at p_u, 1 above","Percolation on tree×lattice: boundary jumps from half to full at p_u","Hausdorff dimension of cluster limit points: ≤1/2 at p_u, 1 above"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the matching lower bound in Lemma 2.4, whose proof is only sketched with the sentence 'we omit further details here' and deferred to the techniques of [18,33]; the paper also relies on the companion result [26] for the exponential tail, so the equality $\\delta_H(p)=1-\\beta_p^*$ stands or falls with those deferred ingredients.","fun_headline_variants_meta":{"raw":{"variants":["Boundary dimension of infinite clusters jumps from ≤1/2 to 1 at p_u","At p_u, cluster boundary traces jump from half to full dimension","Infinite cluster boundary: ≤1/2 at p_u, 1 above","Percolation on tree×lattice: boundary jumps from half to full at p_u","Hausdorff dimension of cluster limit points: ≤1/2 at p_u, 1 above"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000867,"raw_usage":{"total_tokens":3788,"prompt_tokens":1009,"completion_tokens":2779,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":2667}},"tokens_in":625,"tokens_out":2779,"duration_ms":18034,"temperature":1.0,"reasoning_tokens":2667,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:59:45.277261+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a $p<p_u$ and an amenable $H$ for which the boundary limit set of an infinite cluster has Hausdorff dimension strictly larger than $1/2$; Theorem 1.1 says this cannot happen. A concrete route is to run percolation on $T\\times\\mathbb{Z}$ at a numerically sub-$p_u$ parameter, measure the box-counting or Hausdorff dimension of the cluster's accumulation set on $\\partial T$, and compare it with $1-\\beta_p^*$ estimated from the decay of $P_p(n)$; any systematic gap between the measured dimension and $1-\\beta_p^*$ below $p_u$ would falsify Lemma 2.4.","supporting_citations":[{"cited_title":"Percolation at the uniqueness threshold via subgroup relativization","cited_arxiv_id":"2409.12283","evidence_quote":"Supplies the exponential tail bound for fiber intersections and the characterization of $p_u$ via amenability of the fiber subgroup, used in Lemmas 2.1 and 2.6."},{"cited_title":"Hutchcroft","cited_arxiv_id":null,"evidence_quote":"Provides the nonunimodular percolation framework, tilted susceptibilities, and the $p_c<p_t$ theorem that underpin the threshold equalities and continuity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes infinitely many infinite clusters at $p_u$ for $T\\times\\mathbb{Z}$, the result that Theorem 1.1 strengthens."},{"cited_title":"Hutchcroft","cited_arxiv_id":null,"evidence_quote":"Develops the nonunimodular bridge-decomposition argument used to prove $\\chi_{p,1/2}<\\infty$ and hence $p_t=p_u$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives non-uniqueness at $p_u$ for nonamenable products, used in Lemma 2.1's characterization."}],"review_version":1}