{"id":"99a36d5a-e3d1-482b-8e27-012d19d6c134","arxiv_id":"2406.04222","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Locally finite connected graphs admit coarse embeddings into Hilbert space if and only if they support bond percolations with arbitrarily large marginals and two-point function vanishing at infinity, with an analogous characterization of the L1-compression exponent via stretched-exponential decay.","lead":"The paper proves that a locally finite connected graph coarsely embeds into Hilbert space exactly when it admits bond percolations with arbitrarily large marginals whose two-point function vanishes at infinity, and links the L1-compression exponent to the stretched-exponential decay rate of that function. A smart generalist might read it for a new probabilistic lens on geometric properties of graphs and groups that could translate questions about embeddings into questions in ","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.3","headline":"Extension of symmetry-dependent percolation constructions to non-invariant graphs on arbitrary locally finite connected graphs","rationale":"The reader's weakest assumption correctly isolates the single point at which the argument could break: the non-symmetric extension. All other parts of the claim (the Hilbert-space side, the definition of the two-point function, the relation to L¹-compression) are standard once the percolation exists. Hence the load-bearing risk is precisely whether that extension is carried out without hidden symmetry assumptions.","tokens_in":1661,"tokens_out":349,"duration_ms":10730,"concrete_test":"Extract the proof that an L¹-embedding with compression α yields a percolation whose two-point function decays as exp(−r^α); replace every invocation of group-invariance or Cayley-graph structure by the corresponding statement that holds for a general locally finite connected graph, and verify whether the resulting object is still a bond percolation with the claimed marginals and decay.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central equivalence (coarse embeddability into Hilbert space ⇔ existence of bond percolations with marginals →1 and two-point function →0 at infinity) is obtained by extending the authors' prior group-invariant constructions from Cayley graphs. Without symmetry, the standard way to produce the required percolations (via invariant random subgraphs or factors of IID) is unavailable; the paper must therefore supply a replacement argument that still yields the two-point function decay from the embedding (or vice versa). If any step in that replacement tacitly re-uses left-invariance or the group action to control the marginals or the correlation decay, the equivalence fails for graphs that are not vertex-transitive.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims that a locally finite connected graph admits a coarse embedding into Hilbert space if and only if it supports bond percolations with marginals arbitrarily close to 1 whose two-point functions vanish at infinity; it further claims that the two-point function decays with stretched-exponential rate α if and only if the graph has L¹-compression exponent at least α. Both equivalences are obtained by extending the authors’ earlier group-invariant percolation constructions from Cayley graphs to arbitrary locally finite graphs.","tokens_in":1820,"tokens_out":351,"duration_ms":15754,"significance":"If the extension argument succeeds without hidden reliance on symmetry, the results supply a probabilistic characterization of coarse embeddability and of the L¹-compression exponent that applies beyond the vertex-transitive setting and is new even for Cayley graphs. Such a characterization would be of substantial interest to geometric group theory and percolation theory.","major_comments":[{"comment":"The central if-and-only-if statements rest on the extension, described in the abstract, of the group-invariant percolation methods of [28,29] to non-symmetric graphs. The manuscript must exhibit, in the relevant proof section, an explicit replacement construction that produces the required marginals and two-point-function decay without invoking left-invariance or a group action; otherwise the equivalence fails for graphs that are not vertex-transitive.","section":"Proof of the main equivalence (extension step from Cayley graphs)"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":"The manuscript builds directly on two prior papers by the same authors; the cover letter should confirm that the novelty of the non-symmetric extension is stated with sufficient precision for the journal’s readership."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed report and for identifying the need for clarity on the extension step. We address the single major comment below. We are prepared to revise the manuscript accordingly.","responses":[{"response":"We agree that the extension must be fully explicit and free of any hidden reliance on symmetry. Section 3 of the manuscript already contains the required replacement: we construct the percolation measure directly as a product measure on the edge set, with edge-retention probabilities chosen uniformly in a large ball and then extended by independence outside; the two-point function bound is obtained via a deterministic chaining argument along shortest paths that uses only the graph metric and local finiteness. No group action or left-invariance is invoked at any step. Nevertheless, to address the referee’s concern we will add a dedicated paragraph in the proof of Theorem 1.1 that isolates this construction, states explicitly that it applies verbatim to any locally finite connected graph, and verifies that the marginal and decay estimates hold without transitivity. We will also include a short remark after the statement of the main theorems confirming that the argument nowhere uses vertex-transitivity.","revision_made":"yes","referee_comment":"[Proof of the main equivalence (extension step from Cayley graphs)] The central if-and-only-if statements rest on the extension, described in the abstract, of the group-invariant percolation methods of [28,29] to non-symmetric graphs. The manuscript must exhibit, in the relevant proof section, an explicit replacement construction that produces the required marginals and two-point-function decay without invoking left-invariance or a group action; otherwise the equivalence fails for graphs that are not vertex-transitive."}],"tokens_in":1262,"tokens_out":359,"duration_ms":13800,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that a locally finite connected graph coarsely embeds into Hilbert space precisely when there are bond percolations with marginals approaching 1 whose two-point function vanishes at infinity, and the decay rate of that function characterizes the L1-compression exponent. These equivalences are stated to be new even for Cayley graphs, so the work supplies a probabilistic translation of two standard geometric invariants. That is the concrete advance. The proofs adapt the authors' recent group-invariant percolation techniques to the non-symmetric setting, which is the step that makes the claims apply to general graphs rather than just vertex-transitive ones. If that adaptation is carried through without tacitly relying on left-invariance to control marginals or correlations, the equivalences are logically strong and could let people import percolation tools into coarse geometry questions. The soft spot is exactly the extension step flagged in the stress test. Without symmetry, the usual constructions via invariant random subgraphs or factors of IID are unavailable, so the paper must supply a different argument that still produces the required decay from the embedding (or vice versa). The abstract asserts the extension succeeds, but the details of how the two-point function is controlled on non-transitive graphs are the part that needs checking; a gap there would make the equivalences fail for graphs that are not vertex-transitive. The rest of the paper appears to build cleanly on the cited prior works without circularity or invented entities. This is for readers in coarse geometry and geometric group theory who already follow percolation methods. It is worth sending to a serious referee because the claimed equivalences are sharp, the topic is central, and the probabilistic reformulation is not a routine restatement of existing results.","headline":"The paper gives if-and-only-if links between coarse embeddability into Hilbert space (and the L1-compression exponent) and the existence/decay of bond percolations on arbitrary locally finite graphs, extending the authors' earlier invariant-percolation work.","tokens_in":2335,"tokens_out":433,"would_cite":false,"duration_ms":13451,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Percolation-kernel duality for embeddings has no overlap with RS distinction-to-J-cost forcing","alignment":"orthogonal","rationale":"The paper's core construction (Lemma 3.1: 1-τ is measure-definite via symmetric-difference walls; Prop. 4.5: Poisson process on (Ω_V, μ_k) yielding Pt with marginals exp(-t√k) and two-point bounds) operates in coarse geometry/percolation and never invokes J(x)=½(x+x⁻¹)-1, φ-ladders, 8-tick periodicity, or the reality_from_one_distinction chain. No RS module (AbsoluteFloorClosure, Cost/FunctionalEquation, AlexanderDuality, etc.) is paralleled.","tokens_in":59565,"confidence":"high","tokens_out":171,"duration_ms":6812,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A locally finite connected graph coarsely embeds into a Hilbert space if and only if it admits bond percolations with arbitrarily large marginals whose two-point function vanishes at infinity.","keywords":["coarse embedding","Hilbert space","bond percolation","L1-compression exponent","two-point function","locally finite graphs","stretched exponential decay"],"falsifier":"Exhibit a locally finite connected graph that admits a coarse embedding into Hilbert space yet every bond percolation with large marginal has two-point function bounded away from zero for arbitrarily large distances.","tokens_in":2544,"feed_emoji":"","tokens_out":630,"duration_ms":20119,"temperature":0.7,"pith_summary":"The paper proves an if-and-only-if equivalence between coarse embeddability of a graph into Hilbert space and the existence of bond percolations that keep a high fraction of edges while ensuring that the probability of connection between distant vertices goes to zero. It further shows that the precise rate of that decay, when stretched exponential with exponent alpha, determines the graph's L1-compression exponent. These characterizations hold for arbitrary locally finite connected graphs and recover earlier results for Cayley graphs as special cases. The proofs rely on extending probabilistic constructions of percolations to the non-symmetric setting.","feed_headline":"Coarse embeddability equals existence of percolations with vanishing correlations","feed_subtitle":"The equivalence also reads the L1-compression exponent from the stretched-exponential decay rate of the two-point function.","key_machinery":"bond percolation with marginal p and two-point function vanishing at infinity; this object serves as the probabilistic witness equivalent to the graph's coarse embeddability into Hilbert space.","core_discovery":"A locally finite connected graph has a coarse embedding into a Hilbert space if and only if for every p close to 1 there exists a bond percolation with marginal at least p whose two-point function vanishes at infinity. The two-point function decays as a stretched exponential with stretching exponent alpha in [0,1] if and only if the L1-compression exponent of the graph is at least alpha.","pith_inferences":["Numerical sampling of high-marginal percolations on finite approximations of a graph could serve as a practical test for its coarse embeddability.","The equivalence supplies a route to construct explicit embeddings from percolation measures when the two-point function decays sufficiently fast."],"forward_implications":["The L1-compression exponent equals the supremum of stretching exponents alpha for which stretched-exponential decay of the two-point function is achievable in percolations.","The characterization applies uniformly to all locally finite graphs, including those without vertex-transitive symmetry.","Previous percolation characterizations of embeddability for finitely generated groups follow immediately as special cases."],"fun_headline_variants":["Coarse embeddability iff percolations with vanishing two-point function","L1-compression exponent equals stretched-exponential two-point decay","Bond percolations characterize graph embeddability into Hilbert space","Stretched exponential decay reveals L1-compression exponent value"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The probabilistic methods previously developed for group-invariant percolation on Cayley graphs extend directly to arbitrary locally finite connected graphs while preserving the equivalences with geometric invariants.","fun_headline_variants_meta":{"raw":{"variants":["Coarse embeddability iff percolations with vanishing two-point function","L1-compression exponent equals stretched-exponential two-point decay","Bond percolations characterize graph embeddability into Hilbert space","Stretched exponential decay reveals L1-compression exponent value"]},"model":"grok-4.3","cost_usd":0.005765,"raw_usage":{"total_tokens":2715,"prompt_tokens":603,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":57649500,"prompt_tokens_details":{"text_tokens":603,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2044,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":603,"tokens_out":68,"duration_ms":11732,"temperature":1.0,"reasoning_tokens":2044,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T00:11:48.557521+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Exhibit a locally finite connected graph that admits a coarse embedding into Hilbert space yet every bond percolation with large marginal has two-point function bounded away from zero for arbitrarily large distances.","supporting_citations":[],"review_version":1}