{"id":"038decdd-4cb7-4515-92bc-020b651027c2","arxiv_id":"2607.22908","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"For an epidemic on a zero-range process, the authors prove a non-trivial critical density for extinction versus survival at every healing rate, and a critical healing threshold at low densities.","lead":"This paper proves that an infection spreading through a zero-range particle system on the integer lattice has a phase transition: at low particle density it dies out, and at high density it survives, for any fixed healing rate. It also proves survival at every positive density when healing is slow, establishing a full extinction-survival phase diagram for this interacting particle system.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Survival theorems rely on Lemma 6.16, which replaces N interacting ZRP messengers by N independent random walks and asserts independence across them without a coupling; this is the weakest load-bearing step.","rationale":"The reader's conditional verdict is appropriate. The decoupling inequalities and extinction proof are detailed and represent real technical progress; the independent support includes explicit proofs of Theorems 1.6 and 1.7 and a full multi-scale triggering argument for extinction. The gap in Lemma 6.16 is exactly where the survival argument's base scale is controlled, and the missing coupling cannot be dismissed as a typo because the independence of the tilde A events is essential for the exponential-in-N bound (6.35). I agree with the reader that this does not imply the theorems are false; it means the manuscript as written has an unproved load-bearing step. A second, related gap is the unproved decreasing-function version of Theorem 1.7 invoked in Lemma 6.9 for the survival recursion; it is not fixed by the present write-up and should also be supplied before the survival theorems are accepted. My recommendation is unchanged: conditional acceptance pending a rigorous proof of Lemma 6.16 (and the decreasing vertical-decoupling statement used in Lemma 6.9).","tokens_in":49968,"tokens_out":23894,"duration_ms":225055,"concrete_test":"Demand an explicit construction for the coupling in Lemma 6.16: on one probability space, give the slot-priority process with N tagged messengers and independent (Y^(m), Pi-_{m}, Pi+_{m})_{m<=N} such that (6.32) holds for every messenger and the events tilde A_{m,i} from (6.34) are independent. If the construction cannot be written, recompute Lemma 6.17 with the Hoeffding step (6.35) replaced by a union bound over messengers; if the resulting upper bound is not O(L0^{3d} e^{-c rho0 L0^{1/4}}), then the base-scale triggering for Theorems 1.2 and 1.4 is not established and the phase transition Theorem 1.3 lacks proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The survival side of the phase diagram (Theorems 1.2, 1.4, and hence 1.3) reduces at base scale to the seeding bound Lemma 6.17, whose stochastic content is Lemma 6.16. The proof after (6.31) asserts that one can assign to each green messenger an independent simple random walk Y^(m) and independent Poisson processes Pi- and Pi+ with rates Gamma- and Gamma+, 'such that Pi- is a thinning of Pi+' and the actual number of jumps J_m satisfies J^-_m <= J_m <= J^+_m. It then uses the independence of the random walks and jump bounds to conclude that the events tilde A_{m,i} are independent, which justifies the Hoeffding bound (6.35). Two things are missing. First, no coupling is exhibited between the tagged-particle trajectory in the zero-range process and the independent random walk; the individual jump rate of a ZRP particle is bounded by Gamma- and Gamma+, but the jump process is driven by slot clocks that depend on local occupancy and are reassigned on interaction, so the sandwich (6.32) is not a consequence of the rate bounds alone. Second, even conditional on E1, the actual jump counts J_m of distinct messengers are correlated through the shared environment; independence of the auxiliary walks does not imply independence of the actual events tilde A_{m,i}. Without (6.35), the union bound gives only P(no messenger reaches target) <= N(1-p_hit), which has no exp{-c rho0 L0^{1/4}} decay, so p0 <= L0^{-5d} is not obtained. Lemma 6.17 is cited in both Theorem 1.2 (after (6.43)) and Theorem 1.4 (after (6.49)); thus the survival regime is not closed as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies an SIS-type infection process on top of a zero-range process (ZRP) in equilibrium at density ρ on Z^d. Transmission is instantaneous whenever an infected and a healthy particle co-occupy a site, infected particles heal at rate δ when isolated, and the infection starts from the particles at the origin. The main results are: low-density a.s. extinction for every fixed δ (Theorem 1.1); high-density survival with positive probability for every δ, including δ=∞ (Theorem 1.2); existence of a nontrivial critical density ρ_c(δ) (Theorem 1.3); survival at every positive density for sufficiently small δ (Theorem 1.4); and the derived critical healing threshold (Theorem 1.5). The proofs combine a slot-priority graphical representation for the ZRP with new horizontal and vertical decoupling inequalities, and then use multi-scale renormalization: an immunity-regime contraction argument for extinction and an oriented-percolation argument for survival. The survival side depends on a seeding lemma (Lemma 6.17) whose stochastic content is the independent-diffusion lemma (Lemma 6.16).","tokens_in":50364,"tokens_out":11080,"duration_ms":114543,"significance":"If the proof is completed, this would be the first survival-extinction phase transition result for an epidemic spreading in a ZRP environment, extending the Kesten--Sidoravicius framework beyond independent random walks. The paper's horizontal decoupling (Theorem 1.6), uniform-in-density vertical decoupling (Theorem 1.7), and the extinction-side renormalization are substantial contributions in their own right, and the estimates are mostly carried out in a careful and self-contained way. The decoupling theorems are stated in a reusable form and are likely to be useful for other models with ZRP dynamic environments. However, the survival theorems all pass through Lemma 6.16, and the current proof of that lemma contains a load-bearing unsupported independence/coupling assertion. Because Theorems 1.2, 1.4, and hence the survival half of Theorem 1.3 rest on that lemma, the main phase-diagram claim is not yet established as written.","major_comments":[{"comment":"The proof asserts, after Eq. (6.31), that each green messenger can be assigned an independent simple random walk Y^(m) and independent Poisson processes Π^-_m and Π^+_m with rates Γ^- and Γ^+ such that the actual jump count J_m satisfies J^-_m ≤ J_m ≤ J^+_m, and then concludes from the independence of these auxiliary objects that the events tilde A_{m,i} are independent, justifying the Hoeffding bound (6.35). Neither step follows from the preceding construction. In the slot representation, the clock governing a tagged particle is tied to a slot and is reassigned on interactions; a per-particle jump-rate bound Γ^- ≤ individual rate ≤ Γ^+ does not by itself produce a coupling of the tagged ZRP trajectory with an independent random walk with the sandwich property (6.32). Moreover, even conditional on E1, the actual jump counts J_m of distinct messengers are correlated through the shared environment, so independence of the auxiliary walks does not imply independence of the actual events tilde A_{m,i}. Without (6.35), the union bound gives only N(1 - p_hit), which does not yield the exp{-c ρ̄0 L0^{1/4}} decay needed for (6.31). Since Lemma 6.17 is used in Eq. (6.43) for Theorem 1.2 and in Eq. (6.49) for Theorem 1.4, this missing coupling is load-bearing for the survival side of the phase diagram.","section":"Section 6.3, proof of Lemma 6.16, Eqs. (6.31)--(6.35)"},{"comment":"The vertical decoupling proof couples two ZRP evolutions by declaring that matched particles at the same site 'jump together' and then applies Proposition 2.12 to their relative displacement, asserting that the relative motion is a continuous-time random walk with jump rate at least 2Γ^-. This is not justified: in the ZRP, when two tagged particles occupy the same site, only one particle jumps at rate g(n), and the motion of each tagged particle depends on the local occupancies; the 'jump together' rule appears to modify the single-particle jump mechanism of the ZRP generator rather than to couple two copies of it. The bounded-below-rate condition needed for Proposition 2.12 is therefore not established. This issue affects the bound of the third term in Eq. (4.13), and hence the proof of Theorem 1.7. Since Theorem 1.7 is invoked in the recursive estimates (5.16) and (6.15) on both the extinction and survival sides, the vertical decoupling theorem is another load-bearing point that needs a rigorous argument.","section":"Section 4.2, matching coupling in the proof of Theorem 1.7"}],"minor_comments":[{"comment":"There is a typographical error in the display: 'P((ξ,ξ′)∈D^c)' contains an extra parenthesis; it should read P((ξ,ξ′)∈D^c) or, more simply, P(D^c).","section":"Section 4.3, Eq. (4.17)"},{"comment":"The symbol ρ∞ is used for the limiting density ℓ0^{-s} in the immunity regime, while the same symbol later denotes a target density in the contagion regime; since ℓ0 is ultimately chosen as a large constant, the notation is confusing and should be adjusted to avoid an apparent circular dependence.","section":"Section 5, Eqs. (5.9)--(5.11)"},{"comment":"In the recruitment argument, the stochastic domination in Eq. (6.28) includes the indicator of Elocal ∩ Edense, but the stopping rule is described as 'abort' when either event fails; the reader must check that the strong Markov property is applied at stopping times that do not depend on future coin flips. A short formal sentence specifying the stopping times would improve readability.","section":"Section 6.3, proof of Lemma 6.15"},{"comment":"The sentence 'By translation invariance and stationarity, the probability that an infinite genealogical path starts from (0,0) is therefore positive' is imprecise: the a.s. chain constructed in Theorem 6.11 is already rooted at the origin, so the desired positivity follows directly from intersecting that almost-sure event with the positive-probability event that the origin is initially occupied; the appeal to translation invariance is unnecessary and potentially misleading.","section":"Remark 6.12"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a substantial amount of correct and useful material, especially the horizontal decoupling, the extinction side, and the oriented-percolation framework. The main obstacle to publication is the unproved coupling and independence assertion in Lemma 6.16, which supports both survival theorems and therefore the central phase-transition claim. If the authors can supply a rigorous coupling for the independent-diffusion lemma, or replace the Hoeffding bound with a valid dependence-tolerant estimate, the paper may well be acceptable after revision. The vertical decoupling matching coupling in Section 4.2 also needs clarification or proof, since the current 'jump together' rule does not transparently preserve the ZRP generator."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Serious paper, worth a referee's time. The core content is new: Theorems 1.1–1.5 give the first extinction–survival phase diagram for an infection spreading on a zero-range process, and the uniform-in-density decoupling inequalities (Theorems 1.6 and 1.7) are reusable tools that go beyond [7]. The extinction side (Theorem 1.1) is detailed and self-contained; the renormalization recursions and the decoupling proofs are careful. I see no circularity and the critical density is obtained by monotonicity, not by fitting. The citation pattern is healthy: self-references are background/technique, not inputs.\n\nThe soft spot is real and it is exactly where the stress test points: Lemma 6.16. The proof asserts a coupling in which each green messenger is assigned an independent random walk and independent Poisson clocks, with the actual jump count sandwiched between the two Poisson counts. That sandwich is not a consequence of the rate bounds alone. In the slot representation a tagged particle's jump intensity is driven by slot clocks that are reassigned on interaction, so its jump process is a state-dependent time change; you need an explicit coupling, and none is given. More importantly, even if each messenger's jump count could be sandwiched, the events tilde A_{m,i} are defined in terms of the auxiliary random walks, and the paper does not show that tilde A_{m,i} implies the actual event A_{m,i}. The independence of the auxiliary walks therefore does not give independence of the events that matter, and the Hoeffding bound (6.35) does not follow. Without that exponential decay, the union bound only gives N(1−p_hit), which cannot produce p0 ≤ L0^{−5d}. Both Theorem 1.2 (after (6.43)) and Theorem 1.4 (after (6.49)) lean on Lemma 6.17, so the survival regime is not closed as written.\n\nEverything else in the paper looks credible to me, and I would be surprised if the main theorems are false. But this is a proof gap in a load-bearing step, not a cosmetic issue. My recommendation: send it to a serious referee, and ask the authors to replace the asserted coupling in Lemma 6.16 with an explicit construction or a rigorous domination argument. Until then, the survival theorems are conditional.","headline":"Genuine first phase diagram for infection in a ZRP, with a real gap in the seeding lemma that blocks the survival theorems as written.","tokens_in":50939,"tokens_out":4182,"would_cite":true,"duration_ms":38589,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60K37"],"pacs":[],"model":"deepseek-v4-flash","headline":"In a zero-range crowd of particles, an infection that spreads on contact undergoes a sharp extinction-to-survival phase transition as particle density crosses a critical value, for every recovery rate including infinitely fast healing.","keywords":["infection processes","zero-range process","phase transition","critical density","healing rate","multi-scale renormalization","decoupling inequalities","oriented percolation"],"falsifier":"In one dimension with the linear rate function g(n)=n (where the zero-range process is simply a system of independent random walks), simulate the epidemic for a fixed healing rate such as δ=1 on a large periodic lattice, and estimate the survival probability over very long time windows across a range of densities: if the extinction/survival boundary vanishes, moves with system size, or sits at density 0 or ∞, the phase-transition claim fails. Alternatively, compare the measured probability that a seeded target box is hit in the full interacting model with the independent-walk estimate used in Lemma 6.16; a systematic and persistent gap would invalidate the seeding domination.","tokens_in":49764,"feed_emoji":"🦠","tokens_out":9291,"duration_ms":77998,"temperature":0.7,"pith_summary":"The paper proves that an infection spreading on contact inside a moving crowd of particles—the zero-range process—has a phase transition in density: for any fixed healing rate δ, there is a critical density ρc(δ) below which the infection dies out almost surely and above which it survives with positive probability. This holds even when healing is instantaneous, provided the crowd is dense enough. It also shows that at any positive density, a sufficiently slow healing rate still allows survival. The results cover the full (ρ, δ) phase diagram and establish non-trivial critical thresholds in both directions.","feed_headline":"An epidemic in a jostling crowd flips at a critical density","feed_subtitle":"Even instantaneous healing cannot stop it once density crosses the threshold, while low density guarantees extinction.","key_machinery":"The load-bearing machinery is a pair of space-time decoupling inequalities for the zero-range process: a horizontal (spatial-separation) covariance bound proved via a slot-priority representation in which high-priority particles ignore low-priority ones, and a vertical (temporal-separation) decoupling with sprinkling proved by coupling the density-ρ process to one at density ρ(1+ε) and matching particles block by block. Around these, the paper builds a multi-scale renormalization: for extinction, 'bad' half-crossing events cascade into two well-separated smaller bad events, yielding a recursion pk+1≤Cpk²+error; for survival, good space-time blocks form an oriented percolation cluster whose failure probabilities contract at the same quadratic rate, triggered by a seeding lemma that turns a local accumulation of infected particles into successful crossings of base-scale blocks.","core_discovery":"Starting from a single infected origin in an equilibrium zero-range process of density ρ, with infection transmitted instantaneously on contact and recovery effective only at isolated particles, the paper establishes an extinction–survival dichotomy. It proves low-density extinction and high-density survival for every recovery rate δ, including δ=∞ (instantaneous healing), and concludes that for each fixed δ there is a finite positive critical density ρc(δ) separating almost-sure extinction from survival with positive probability. It further proves that every positive density supports survival once the healing rate is small enough, and that below the density threshold ρc(∞−) there is a well-defined critical healing rate δc(ρ). The argument rests on new space-time decoupling inequalities for the zero-range process whose constants are uniform in density, embedded in a multi-scale renormalization scheme.","pith_inferences":["If the seeding domination can be replaced by a rigorous coupling, the critical density should be approximable from single-particle hitting data, turning the phase-transition theorem into a quantitative prediction.","The same two decoupling inequalities, being density-uniform, are likely to yield analogous extinction-survival phase transitions for the symmetric exclusion process and other conservative particle systems; testing that would separate the zero-range-specific content from the general mechanism.","The model's rule that recovery is overridden whenever another particle is present is what permits survival at infinite healing; adopting the alternative rule from the independent-walk literature (where healing can occur even in groups) would plausibly destroy the high-density survival theorem, so the phase diagram is genuinely sensitive to this microscopic detail."],"forward_implications":["For every healing rate, including instantaneous healing, the epidemic on the zero-range process has a well-defined critical density, so the interacting system and the independent-walk system studied earlier share the same qualitative phase diagram.","Above a fixed finite density the infection survives regardless of how fast particles heal, because crowded sites are almost never isolated; the survival region is therefore unbounded in the healing-rate direction.","At any positive density the healing rate can be made small enough to guarantee survival, so the extinction region in the (ρ, δ)-plane is bounded below by a positive critical healing curve.","The density-uniform decoupling inequalities are stated for general conservative particle systems, so the same renormalization machinery should transfer to other environments such as the symmetric exclusion process.","The main open questions—whether the critical curve is continuous and strictly increasing, whether the two infinite-healing thresholds coincide, and whether a shape theorem holds—are now explicitly posed and accessible to the same methods."],"supporting_citations":[{"why":"Supplies the invariant product measures µρ that serve as the equilibrium environment on which the infection evolves.","marker":"[2]"},{"why":"Establishes local and global survival for infections with recovery in the independent-walk environment, the result that Theorem 1.2 generalizes to the zero-range setting.","marker":"[5]"},{"why":"Introduces the coupling strategy (for the exclusion process) that the vertical decoupling proof generalizes and refines.","marker":"[6]"},{"why":"Provides the zero-range-process slot representation and prior decoupling bounds that the horizontal decoupling (Theorem 1.6) extends and improves.","marker":"[7]"},{"why":"Initiates the multi-scale renormalization with half-crossing events that the extinction proof (Theorem 1.1) adapts to the zero-range environment.","marker":"[14]"},{"why":"Supplies the cascading property for half-crossings in contact processes with dynamic environments, used to derive the quadratic recursion for extinction.","marker":"[19]"},{"why":"Defines the original moving-population infection model whose phase diagram this paper extends from independent walks to the interacting zero-range process.","marker":"[22]"},{"why":"Proves the healing-rate phase transition for the independent-walk model; its microscopic transmission rule is the contrast that motivates the present instantaneous-transmission rule.","marker":"[23]"}],"fun_headline_variants":["Zero-range epidemic: density flips extinction to survival","Critical density marks epidemic survival in zero-range process","Even instant healing can't stop zero-range epidemic at high density","Density threshold determines epidemic extinction or survival in zero-range process"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's weakest spot is the seeding lemma's assumption that infected 'messenger' particles act as independent random walks with bounded jump rates, with a hit in that independent picture guaranteeing a genuine infection path in the interacting zero-range process.","fun_headline_variants_meta":{"raw":{"variants":["Zero-range epidemic: density flips extinction to survival","Critical density marks epidemic survival in zero-range process","Even instant healing can't stop zero-range epidemic at high density","Density threshold determines epidemic extinction or survival in zero-range process"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000466,"raw_usage":{"total_tokens":2293,"prompt_tokens":878,"completion_tokens":1415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":1349}},"tokens_in":494,"tokens_out":1415,"duration_ms":9455,"temperature":1.0,"reasoning_tokens":1349,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:28:25.338592+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In one dimension with the linear rate function g(n)=n (where the zero-range process is simply a system of independent random walks), simulate the epidemic for a fixed healing rate such as δ=1 on a large periodic lattice, and estimate the survival probability over very long time windows across a range of densities: if the extinction/survival boundary vanishes, moves with system size, or sits at density 0 or ∞, the phase-transition claim fails. Alternatively, compare the measured probability that a seeded target box is hit in the full interacting model with the independent-walk estimate used in Lemma 6.16; a systematic and persistent gap would invalidate the seeding domination.","supporting_citations":[{"cited_title":"Invariant Measures for the Zero Range Process.The Annals of Probability, 10(3):525 – 547, 1982","cited_arxiv_id":null,"evidence_quote":"Supplies the invariant product measures µρ that serve as the equilibrium environment on which the infection evolves."},{"cited_title":"Local and global survival for infections with recovery.Stochastic Processes and their Applications, 160:161–173, 2023","cited_arxiv_id":null,"evidence_quote":"Establishes local and global survival for infections with recovery in the independent-walk environment, the result that Theorem 1.2 generalizes to the zero-range setting."},{"cited_title":"How can a clairvoyant particle escape the exclusion process?Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques, 54(4):2177–2202, 2018","cited_arxiv_id":null,"evidence_quote":"Introduces the coupling strategy (for the exclusion process) that the vertical decoupling proof generalizes and refines."},{"cited_title":"Spread of an infection on the zero range process","cited_arxiv_id":null,"evidence_quote":"Provides the zero-range-process slot representation and prior decoupling bounds that the horizontal decoupling (Theorem 1.6) extends and improves."},{"cited_title":"Mountford, Daniel Ungaretti, and Maria Eul´ alia Vares","cited_arxiv_id":null,"evidence_quote":"Initiates the multi-scale renormalization with half-crossing events that the extinction proof (Theorem 1.1) adapts to the zero-range environment."},{"cited_title":"Results on the contact process with dynamic edges or under renewals.Electronic Journal of Probability, 27(none):1 – 31, 2022","cited_arxiv_id":null,"evidence_quote":"Supplies the cascading property for half-crossings in contact processes with dynamic environments, used to derive the quadratic recursion for extinction."},{"cited_title":"The spread of a rumor or infection in a moving population.Annals of Probability, pages 2402–2462, 2005","cited_arxiv_id":null,"evidence_quote":"Defines the original moving-population infection model whose phase diagram this paper extends from independent walks to the interacting zero-range process."},{"cited_title":"A phase transition in a model for the spread of an infection.Illinois Journal of Mathematics, 50(1-4):547–634, 2006","cited_arxiv_id":null,"evidence_quote":"Proves the healing-rate phase transition for the independent-walk model; its microscopic transmission rule is the contrast that motivates the present instantaneous-transmission rule."}],"review_version":2}