{"id":"8024e5aa-7a70-4e16-8d4d-45adfb8623a2","arxiv_id":"2505.07145","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For worms on any nonamenable unimodular transitive graph and for arbitrary animals on nonamenable free products, infinite second moment forces infinite clusters at every positive intensity.","lead":"This probability paper studies the Poisson zoo, a model where random connected clusters are dropped independently on each vertex of an infinite graph. It claims that on nonamenable graphs, even at arbitrarily low density, rare huge clusters can connect into infinite components for worms and for general animals on free products.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 3.9 rests on an inequality, (3.27), that is false as written: a worm hitting several vertices of B is counted once on the left but once per hit on the right. Since this is the key first-moment input for Theorem 1.1, the main worm result is not established as written.","rationale":"Reading the paper in good faith, the central achievements are the two λ_c=0 theorems and the uniqueness example. The free-product theorem is proved through Corollary 3.7 and a branching exploration that do not use the disputed inequality, and Proposition 1.4 is an independent construction. The worm theorem, however, is proved through the chain Corollary 3.9 → Proposition 4.8 → Theorem 2.17 → Theorem 1.1, and Corollary 3.9 is the only place where the first moment of the fat is lower-bounded linearly in |A| for general nonamenable graphs. The reader's weakest_assumption identifies precisely the point where the argument fails. Checking Definition 3.1 confirms the overcounting: the restriction defining ˇX^R_{A,{x}} excludes animals that hit A\\{x}, but a vertex y∈B with y≠x is not in A, so an animal may hit several vertices of B and still belong to every one of the singleton bundles. Thus (3.27) has the wrong inequality sign, and the derivation of (3.29) from Corollary 2.9 cannot be made as written. Replacing the false inequality by the correct reversed one would give an upper bound, not the lower bound needed for (4.27). The worm theorem may well be repairable, for instance by discarding or carefully assigning multi-hit animals, but the current manuscript does not supply that repair. Therefore the verdict should remain REJECT for the paper as written, while acknowledging that the free-product result and the uniqueness example are not affected by this specific flaw.","tokens_in":37583,"tokens_out":4990,"duration_ms":52056,"concrete_test":"Exhibit a deterministic counterexample to (3.27): choose any finite connected A, two distinct vertices b1,b2∈∂extA, and a connected animal H with |H|≤R, H∩A=∅, and H∩B={b1,b2}. For the point measure X=δ_{(x,H)}, Definition 3.1 gives ˇΣ^R_{A,B}=|H| whereas ˇΣ^R_{A,{b1}}+ˇΣ^R_{A,{b2}}=2|H|, so (3.27) fails. Then, independently of this example, re-derive Corollary 3.9 without using (3.27); if the only proposed fix is to redefine ˇX^R_{A,{x}} so that multi-hit worms are discarded, the resulting lower bound is no longer what is stated in (3.26) and the proof of Theorem 1.1 still needs a new argument.","verdict_should_be":"REJECT","load_bearing_attack":"In Corollary 3.9, the proof of the lower bound (3.26) passes through (3.27), which claims ˇΣ^R_{A,B} ≥ Σ_{x∈B} ˇΣ^R_{A,{x}}. According to Definition 3.1, ˇX^R_{A,{x}} consists of animals of volume at most R that hit {x} and do not intersect A\\{x}. Since B⊆∂extA, this condition does not prevent the same animal from also hitting other vertices of B. Hence an animal with |H∩B|=k is counted once on the left-hand side and k times on the right-hand side, so the displayed inequality is actually reversed. The accompanying sentence says that worms hitting several vertices of B are 'thrown away,' but no such discarding appears in (3.1). Because Corollary 3.9 is the only first-moment lower bound used to verify hypothesis (4.27) of Proposition 4.8, the proof of Theorem 2.17 and hence Theorem 1.1 loses its main engine: without (3.27), Lemma 3.8 cannot be summed over B together with Corollary 2.9 to obtain the linear lower bound in |A|. This is a load-bearing gap in the central worm theorem, even though Theorem 1.2 on free products and Proposition 1.4 do not rest on this step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Poisson zoo model on transitive graphs, in which i.i.d. Poisson(λ) copies of random finite connected subsets (lattice animals) are placed with intensity λ. The three main claims are: (1) for any nonamenable unimodular transitive graph, worms of random length with infinite second moment have λ_c=0 (Theorem 1.1, proved as Theorem 2.17); (2) for any nonamenable free product of transitive graphs and any animal measure with infinite second moment, λ_c=0 (Theorem 1.2); and (3) on T_d×Z^5 there is a zoo with finite first moment and a unique infinite cluster for every λ>0 (Proposition 1.4). The proofs are built on exploration processes: a branching process on free products, and a fattening exploration process on general nonamenable graphs, supported by first- and second-moment bounds on the newly occupied set. I find a load-bearing false inequality in the worm proof, Corollary 3.9, inequality (3.27), and an additional invalid algebraic step in the same corollary. As written, the proof of Theorem 1.1 is not established.","tokens_in":37864,"tokens_out":17235,"duration_ms":175857,"significance":"If correct, the results would be a substantial advance: they resolve the worm case for all nonamenable unimodular transitive graphs, give the first general-animal result on free products, and provide an appealing example of immediate uniqueness with a sparse FIID cluster. The paper uses elegant tools, including size-biasing via unimodular mass transport, capacity bounds on nonamenable graphs, and Poisson point process restrictions; these parts are mostly well executed. However, the worm theorem is one of the two headline results, and its proof currently rests on a false inequality and an incorrect algebraic passage. The free-product theorem and the uniqueness example are independent of the worm argument, but the manuscript as a whole cannot be accepted until the worm proof is repaired.","major_comments":[{"comment":"Inequality (3.27) is false as written. By Definition 3.1, the left-hand side counts each animal of volume at most R that hits B and avoids A\\B exactly once, while each summand on the right counts the same animal once for every vertex of B it hits. For an animal whose trace intersects B in k vertices, the contribution to the right-hand side is k times its contribution to the left-hand side, so the correct inequality is the reverse, not (3.27). The sentence in the proof saying that worms hitting several vertices of B are \"thrown away\" does not correspond to any restriction present in (3.1). This is load-bearing: Corollary 3.9 is the only first-moment lower bound used to verify hypothesis (4.27) of Proposition 4.8, and without (3.27) the sum over B of the single-vertex bounds from Lemma 3.8 does not yield a lower bound on the total occupation measure of the unrestricted process. Thus the proof of Theorem 2.17, and hence of Theorem 1.1, collapses at this point unless a correct argument replacing (3.27) is supplied.","section":"§3.3, Corollary 3.9, Eq. (3.27)"},{"comment":"The displayed chain in (3.29) is also algebraically invalid. From (2.20) one gets Σ p_x(p_x−ε) ≥ (1−ρ)^2|A| − (|∂extA|−|B|) − ε|A|. The next line claims this is at least ((1−ρ)^2−ε)(1+h)|A| − (|∂extA|−|B|), but the difference between the claimed lower bound and the obtained bound is h[(1−ρ)^2−ε]|A|, which is positive for h>0 and ε<(1−ρ)^2. Therefore the factor (1+h) in (3.29), and consequently the factor (1+h)/2 in (3.26), is not derived from Corollary 2.9. For graphs with h>1, this factor asks for a strictly stronger capacity lower bound than Lemma 2.8 provides. The constants in (3.26) and in hypothesis (4.27) therefore need to be reworked independently of the issue in (3.27).","section":"§3.3, Corollary 3.9, Eq. (3.29)"},{"comment":"The proof of Proposition 1.4 relies on the in-preparation reference [23] for indistinguishability of infinite clusters in Poisson zoos. Since no proof of this statement appears in the manuscript, the proposition is currently conditional on an external unpublished result. The authors should either include a proof of the needed indistinguishability statement, give a publicly available reference, or explicitly mark Proposition 1.4 as conditional.","section":"§5, Proposition 1.4"}],"minor_comments":[{"comment":"The condition H∩A={x} in (3.19) and (3.21) should be H∩A=∅, since x∈∂extA and hence x∉A. As written, the event is empty, although the subsequent estimates clearly use the intended avoidance condition.","section":"§3.3, Lemma 3.8, Eqs. (3.19) and (3.21)"},{"comment":"The theorem states only E[L^2]=∞, while the abstract and Lemma 2.16 also impose E[L]<∞. The missing condition should be stated explicitly, even though the case E[L]=∞ is trivial by Lemma 2.14.","section":"§2.5, Theorem 2.17"},{"comment":"There is a typo \"substitue\" in the sentence before (3.25); it should be \"substitute.\"","section":"§3.3, proof of Lemma 3.8"}],"recommendation":"major_revision","confidential_remarks":"The worm theorem, which is one of the two headline results, is not proved as written because of the false inequality (3.27) and the invalid passage (3.29). I would not recommend acceptance in the current form. The free-product theorem appears substantially developed and may be salvageable, and the uniqueness example is attractive, but the unresolved dependence on [23] should also be addressed. A revised version that supplies a correct first-moment lower bound for the worm exploration and fixes the constants would be worth reconsidering."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has two genuinely new theorems and one nice example. Theorem 1.2 (general animals on free products) and Proposition 1.4 (immediate uniqueness on T_d x Z^5) look serious and, as far as I can check, correct. The proof of Theorem 1.2 uses a sprinkling argument to handle the parity issue; the branching process construction is elaborate but plausible. The uniqueness example is a clean application of Burton-Keane plus indistinguishability.\n\nThe problem is Theorem 1.1, stated for worms on any nonamenable unimodular transitive graph. The proof runs through Corollary 3.9, whose key step is inequality (3.27): it claims the total occupation measure of animals hitting B while avoiding A is at least the sum over the corresponding measures for individual vertices of B. With Definition 3.1, an animal of volume at most R that hits B and avoids A is counted once on the left, but on the right it is counted once for each vertex x in B it hits. So the right side is the larger one, and the inequality is simply backwards. The accompanying sentence says worms hitting several boundary vertices are 'thrown away,' but no such discarding appears in the definition of the restricted process. This is not a cosmetic slip: (3.27) is the only route to the first-moment lower bound (3.26), which is exactly hypothesis (4.27) of Proposition 4.8. Without it, the exponential-growth argument for the worm exploration has no fuel. So Theorem 1.1 is not established as written.\n\nThe gap may well be repairable: one can imagine defining a restricted process that genuinely discards multi-hit worms, or replacing (3.27) by a more careful first-moment calculation. The free-product theorem and Proposition 1.4 do not depend on this step. But as it stands, the advertised headline result has a load-bearing false inequality.\n\nThis is a paper for a serious specialist in percolation and random walks. I would send it to a good referee, because the free-product result alone is probably worth publishing, and the worm theorem is close enough that a fix might work. I would not accept Theorem 1.1 in the current form.","headline":"Strong new results for free products and a nice uniqueness example, but the worm theorem on general nonamenable graphs has a false inequality at (3.27) that currently sinks Theorem 1.1.","tokens_in":38427,"tokens_out":2114,"would_cite":false,"duration_ms":20864,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B41","37A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that on nonamenable unimodular transitive graphs, random-walk worms with infinite squared length percolate at every positive intensity, and that the same holds for arbitrary lattice animals on nonamenable free products.","keywords":["Poisson zoo","lattice animals","percolation","nonamenable graphs","random walk worms","free products","phase transition","unimodularity"],"falsifier":"Compute both sides of inequality (3.27) for a worm process on a regular tree, taking $A$ to be a short path and $B$ to be two adjacent exterior vertices, with a worm length distribution that lets one worm touch both vertices. If the left side is strictly smaller than the right side for some $R$, the stated lower bound lacks its proof; if a corrected argument restores the inequality, the conclusion of the worm theorem follows. A direct counterexample to the theorem would be a nonamenable unimodular transitive graph and worm lengths with $\\mathbb{E}L^2=\\infty$ but $\\lambda_c>0$.","tokens_in":37374,"feed_emoji":"🐛","tokens_out":8993,"duration_ms":82513,"temperature":0.7,"pith_summary":"This paper tries to establish that the Poisson zoo can percolate at arbitrarily small intensity on nonamenable transitive graphs, provided the random lattice animals have infinite expected squared size but finite expected size. For random-walk worms this is proved on every nonamenable unimodular transitive graph; for arbitrary animals it is proved on nonamenable unimodular free products. The interest is that these models have infinite clusters despite arbitrarily low density, a behaviour impossible for Bernoulli percolation, and they give sparse factor-of-i.i.d. percolations relevant to measurable cost. A companion example on $T_d\\times\\mathbb{Z}^5$ has a unique infinite cluster at every positive intensity, with finite first moment. The fully general case -- arbitrary animals on arbitrary nonamenable Cayley graphs -- is left open.","feed_headline":"Nonamenable graphs: infinite second moment percolates at any density","feed_subtitle":"Worms with infinite squared length percolate on any nonamenable unimodular graph, even at arbitrarily low density.","key_machinery":"The carrying mechanism is an exploration process that 'fattens' the already explored cluster through its exterior boundary, adding only animals that touch the boundary and avoid the old set. For worms, the expected size of the fat is controlled by random-walk capacity, which is comparable to volume on nonamenable graphs; unimodularity produces size-biased first moments, a corollary neglects multiplicities, and a first-moment lower bound combined with a second-moment upper bound feeds a concentration lemma proving exponential growth forever. For free products, the exploration embeds a Galton-Watson branching process using the fact that every vertex is a cutpoint, with a sprinkling step that repairs the parity issue when growth is available in only one factor of the product.","core_discovery":"The paper's central claim is Theorem 1.1: for any nonamenable unimodular transitive graph $G$, the worms model -- animals that are traces of simple random walks of random finite length $L$ -- has $\\lambda_c=0$ whenever $\\mathbb{E}L<\\infty$ and $\\mathbb{E}L^2=\\infty$. Theorem 1.2 extends the same conclusion to arbitrary lattice animal measures with $\\mathbb{E}|H|^2=\\infty$ when $G$ is a nonamenable unimodular free product. Proposition 1.4 constructs a zoo on $\\mathbb{T}_d\\times\\mathbb{Z}^5$ with $\\mathbb{E}|H|<\\infty$ and a unique infinite cluster for every $\\lambda>0$. The paper leaves open whether $\\lambda_c=0$ holds for every animal measure with infinite second moment on every nonamenable Cayley graph.","pith_inferences":["A testable extension suggested by the method is that worms with infinite second moment may also have $\\lambda_c=0$ on other transient graphs where capacity is not linear, such as high-dimensional amenable lattices; the linear-capacity proof would not transfer directly.","The concentration lemma that turns bounded variance into survival is a transferable tool: any Poisson soup whose growth increments satisfy comparable first and second moment bounds should exhibit permanent supercriticality.","The uniqueness example points toward constructing sparse factor-of-i.i.d. percolations with a unique infinite cluster on groups with vanishing first $\\ell^2$-Betti number, which by known criteria would imply cost $1$; the paper raises this as Question 1.5 but does not resolve it.","If the disputed fattening inequality is repaired, the worm theorem would be proven; if not, the theorem may still be true but would need a different exploration bound."],"forward_implications":["If Theorem 1.1 is correct, every nonamenable unimodular transitive graph admits worm zoos with $\\lambda_c=0$ for every length distribution with finite mean and infinite second moment.","If Theorem 1.2 is correct, on nonamenable free products the critical intensity is zero for every animal measure with infinite second moment, independent of animal shape.","Arbitrarily sparse Poisson zoos can percolate on nonamenable graphs, so in this model low density does not prevent infinite clusters.","The $\\mathbb{T}_d\\times\\mathbb{Z}^5$ example shows $\\lambda_u=0$ with finite first moment: uniqueness of the infinite cluster can also be immediate.","The general Question 1.3 -- arbitrary animals on arbitrary nonamenable Cayley graphs -- remains open."],"supporting_citations":[{"why":"Introduces the Poisson zoo, proves the basic phase-transition facts used here, and establishes the Z^d worm result that Theorem 1.1 extends.","marker":"[46]"},{"why":"Supplies the standard facts on nonamenability, spectral radius, unimodularity, mass transport, and branching process survival used throughout.","marker":"[38]"},{"why":"Provides the random walk and capacity estimates, including linear speed and capacity comparisons, cited in the worm argument.","marker":"[43]"},{"why":"Establishes the mass transport principle for invariant percolation that underlies the size-biasing lemma.","marker":"[5]"},{"why":"Gives the capacity linearity estimate used to control visibility of explored sets in the worm exploration.","marker":"[49]"},{"why":"Provides an alternative source for the linearity of capacity on nonamenable graphs.","marker":"[7]"},{"why":"Shows indistinguishability of infinite clusters for Poisson zoos, used in the uniqueness example.","marker":"[23]"}],"fun_headline_variants":["Infinite second moment makes worms percolate on any nonamenable graph","Heavy-tailed animals force infinite clusters even at arbitrarily low density","Nonamenable free products: infinite variance forces percolation for any animals","Worms with infinite squared length percolate at any rate on nonamenable graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The worm proof assumes that worms touching several boundary vertices can be counted vertex-by-vertex without loss, so that the total fat is at least the sum of single-vertex fats; the displayed inequality does this, but the definitions do not discard multi-touching worms.","fun_headline_variants_meta":{"raw":{"variants":["Infinite second moment makes worms percolate on any nonamenable graph","Heavy-tailed animals force infinite clusters even at arbitrarily low density","Nonamenable free products: infinite variance forces percolation for any animals","Worms with infinite squared length percolate at any rate on nonamenable graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2883,"prompt_tokens":974,"completion_tokens":1909,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":1827}},"tokens_in":590,"tokens_out":1909,"duration_ms":14693,"temperature":1.0,"reasoning_tokens":1827,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:25:22.899683+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of inequality (3.27) for a worm process on a regular tree, taking $A$ to be a short path and $B$ to be two adjacent exterior vertices, with a worm length distribution that lets one worm touch both vertices. If the left side is strictly smaller than the right side for some $R$, the stated lower bound lacks its proof; if a corrected argument restores the inequality, the conclusion of the worm theorem follows. A direct counterexample to the theorem would be a nonamenable unimodular transitive graph and worm lengths with $\\mathbb{E}L^2=\\infty$ but $\\lambda_c>0$.","supporting_citations":[{"cited_title":"R´ ath and S","cited_arxiv_id":null,"evidence_quote":"Introduces the Poisson zoo, proves the basic phase-transition facts used here, and establishes the Z^d worm result that Theorem 1.1 extends."},{"cited_title":"Lyons and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the standard facts on nonamenability, spectral radius, unimodularity, mass transport, and branching process survival used throughout."},{"cited_title":"Pete Probability and Geometry on Groups: Lecture notes for a graduate course","cited_arxiv_id":null,"evidence_quote":"Provides the random walk and capacity estimates, including linear speed and capacity comparisons, cited in the worm argument."},{"cited_title":"Benjamini, R","cited_arxiv_id":null,"evidence_quote":"Establishes the mass transport principle for invariant percolation that underlies the size-biasing lemma."},{"cited_title":"Teixeira (2009) Interlacement percolation on transient weighted graphs.Electronic Jour- nal of Probability 14, 1604–1627","cited_arxiv_id":null,"evidence_quote":"Gives the capacity linearity estimate used to control visibility of explored sets in the worm exploration."},{"cited_title":"Benjamini, A","cited_arxiv_id":null,"evidence_quote":"Provides an alternative source for the linearity of capacity on nonamenable graphs."},{"cited_title":"El Alami, G","cited_arxiv_id":null,"evidence_quote":"Shows indistinguishability of infinite clusters for Poisson zoos, used in the uniqueness example."}],"review_version":1}