{"id":"66f27d68-8fbe-47b8-8f38-a69a059b56f9","arxiv_id":"2507.00329","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For oriented bond-site percolation with columnar stretches, a (1+ε)-moment condition on stretches suffices for a percolation phase transition, yielding survival of contact processes with periodic recovery.","lead":"This paper proves that a stretched, randomly deformed version of oriented percolation percolates once the infection rate is large enough, and uses this to show that certain contact processes with periodic recovery survive. It extends earlier phase-transition results to a broader class of columnar random environments.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reader's dependence concern about Case (Uni) does not land: torus distances from an i.i.d. uniform sequence are actually i.i.d. The real issue is that Theorem 1.2's condition (ii) requires the lower bound for every λ≥0, while the paper's examples have κ_0=0 and the proof only treats λ≥2.","rationale":"The reader identified a real statement-level defect in Theorem 1.2's condition (ii), but the other claimed load-bearing issue--dependent stretches in the Case (Uni) coupling--does not survive scrutiny. For i.i.d. uniforms U_i, the torus distances S_i=2d_T(U_i,U_{i+1}) are independent: conditional on U_i, S_i is uniform on [0,1] and U_{i+1} is fresh randomness, so S_i is independent of the past. Hence the OPRE environment built from ν_i=-log S_i is i.i.d. exponential, and Theorem 1.2's independence hypothesis is satisfied in that coupling. The genuine problem is the quantifier in condition (ii): it demands exp(-σs)≤κλ(s) for every λ≥0, but κ_0=0 for the examples, and the proofs only verify the lower bound for λ≥2 (Uni) or λ≥100 (Ber). This is a formal gap between the theorem statement and its applications, though it is easily repaired by requiring the lower bound only for sufficiently large λ. Because the central renormalization argument does not use small λ, the results should be accepted after a correction; the verdict remains conditional.","tokens_in":17642,"tokens_out":13234,"duration_ms":148346,"concrete_test":"Evaluate κ_0(s) for the paper's example κλ(s)=1-exp(-λ e^{-s}) at λ=0: κ_0(s)=0 for every s, so exp(-σs)≤κ_0(s) fails for every σ>0. This settles whether condition (ii) as stated is satisfiable by the paper's own connection functions. If the hypothesis is weakened to 'for all sufficiently large λ', the proof of Lemmas 2.6-2.8 and Theorem 1.2 remains valid, and the main percolation results stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Reader's weakest assumption is not the right one. In Case (Uni), S_i=2d_T(U_i,U_{i+1}). For any fixed u, the torus distance from u to an independent uniform U is uniform on [0,1/2]; hence S_i is uniform on [0,1] and, conditional on U_i, independent of the past. Thus (S_i) is i.i.d. uniform and ν_i=-log(S_i) is i.i.d. exponential, so the OPRE environment is i.i.d. as required. The load-bearing defect is in Theorem 1.2 itself. Condition (ii) states exp(-σs)≤κλ(s) for every s∈N and every λ≥0. For the paper's own example κλ(s)=1-exp(-λ e^{-s}), κ_0(s)=0, so no σ>0 exists. The proof of Theorem 1.6 (Uni) checks the lower bound only for λ≥2 and s≥s0, then handles 1≤s≤s0 by taking λ large; it never establishes the bound for λ<2. The same mismatch appears in Case (Ber), where λ≥100 is assumed. Therefore, as written, the applications do not satisfy the hypotheses of the theorem. The fix is local: replace 'for every λ≥0' in (ii) by 'for every λ≥λ0 for some λ0≥0' (or add a separate condition only for large λ); the renormalization proof only needs the bound for the large-λ regime.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies oriented bond-site percolation on the planar lattice L2 with columnar random stretches (ξ_x) and (ν_{x,x+1}) and a family of connection functions κλ. Theorem 1.2 asserts that, under a (1+ε)-moment condition on the stretches and certain assumptions on κλ, the model percolates for all sufficiently large λ. The proof uses a static multiscale renormalisation scheme adapted from Hilário et al. [12]. The result is applied to contact processes with periodic recovery (continuous and discrete random shifts) and to contact processes in a random spatial environment, yielding survival for large infection rates; the paper also proves absence of percolation for temporal stretches with heavier-than-exponential tails (Proposition 1.3).","tokens_in":17990,"tokens_out":25117,"duration_ms":268719,"significance":"If Theorem 1.2 is taken with the corrected hypothesis on λ indicated below, the paper is a solid contribution: it generalises the renormalisation scheme of [12] from undirected percolation to oriented bond-site percolation, and the couplings to the contact-process applications are explicit and checkable. The moment condition E[ξ^{1+ε}], E[ν^{1+ε}]<∞ is natural, and the applications give non-trivial survival/extinction phase transitions when combined with the extinction result of [7]. The paper contains no fitted parameters or circular predictions; the main technical input from [12] is external and published. Proposition 1.3 is a clean, correct observation. The main defect is in the statement of condition (ii) of Theorem 1.2, which does not match the paper's own examples; this is local and fixable. The dependence concern raised in the stress-test note about Case (Uni) is, in my reading, not an actual problem.","major_comments":[{"comment":"Condition (ii) as stated requires exp(-σs) ≤ κλ(s) for every λ≥0, but this is false for the paper's own examples: in Section 2.1.1, κλ(s)=1-exp(-λe^{-s}) and in Section 2.1.2, κλ(s)=P(A_⌊√s⌋), and both give κ_0(s)=0 for every s, so no σ>0 can exist. The verification in Section 2.1.1 proves the lower bound only for λ≥2 (Eq. (6)) and handles s≤s0 only for sufficiently large λ; Section 2.1.2 explicitly assumes λ≥100. The renormalisation proof (Lemma 2.6 and Section 2.3) needs the lower bound only for the single large λ at which the scheme is run, not uniformly for all λ≥0. I recommend changing condition (ii) to require the exponential lower bound for all λ≥λ0 for some λ0≥0 (equivalently, for all sufficiently large λ), and adjusting the applications to verify this with their respective λ0. This is a local fix and the main proof is unaffected.","section":"Theorem 1.2, Eq. (3)"}],"minor_comments":[{"comment":"The stress-test concern about dependence of the stretches in Case (Uni) does not land: conditionally on U_i, S_i=2d_T(U_i,U_{i+1}) is uniform on [0,1] and independent of (U_0,...,U_{i-1}), and the resulting sequence (S_i) is i.i.d. uniform; hence ν_i=-log S_i forms an i.i.d. exponential environment as required by Theorem 1.2.","section":"Section 2.1.1, Case (Uni)"},{"comment":"The symbol DTC in the definition of q_k just before Lemma 2.6 should be BTC, the bottom-top crossing event defined in the same section; the right-left crossing RLC was defined, but there is no DTC event.","section":"Section 2.2.2, definition of q_k"},{"comment":"In the text 'set ν := K 2', the notation should read ν := K^2; the identity P(A_K)=κλ(ν) is only correct with ν=K^2, since κλ(s)=P(A_⌊√s⌋).","section":"Section 2.1.2, Case (Ber)"},{"comment":"In the lower bound for traversing a bad area, the displayed factors should be e^{-ν_{x,x+1}} e^{-ξ_x}, not e^{ν_{x,x+1}} e^{ξ_x}; as written the inequality is false because κλ(s) is bounded above by 1, not by e^s.","section":"Section 2.2.3, Eq. (18)"},{"comment":"There is a duplicated phrase 'this argument this argument' in the discussion of almost deterministic interarrival times; please correct the typo.","section":"Section 1.2, paragraph on periodic recovery"},{"comment":"Lemma 2.4, the decoupling estimate P(I_{k,i} is bad) ≤ L_k^{-α}, is stated without proof and attributed to [12, Lem. 3.1]; since Theorem 1.2's induction in Lemma 2.6 and the Borel–Cantelli argument at (19) rely on it, please state the decoupling inequality used and explain why it transfers verbatim to the present bond-site setting, or include the proof.","section":"Section 2.2.1, Lemma 2.4"},{"comment":"The sum in the definition of ν (N_0 log N_0 + ∑_{\\ell=0}^{N_0} Δ_\\ell) does not match the exponent in Eq. (10), which sums over \\ell=X_b+1,\\dots,X_{b+1}-1; please make the stochastic domination explicit (for instance by taking ν = N log N + ∑_{i=1}^{N} Δ'_i with independent copies) so that κ_L(ν) ≤ P(edge open) holds.","section":"Section 2.1.3, Proposition 1.7"},{"comment":"In the definition of H_k, the events LRC(R_ver_k(i,i)) and BTC(R_hor_k(i,i+1)) appear interchanged; based on the rectangle definitions (15)–(16), H_k should involve LRC(R_hor_k) and BTC(R_ver_k) for the patching argument in Figure 13 to be correct.","section":"Proof of Theorem 1.2, definition of H_k"}],"recommendation":"major_revision","confidential_remarks":"The main issue is a mismatch between the statement of Theorem 1.2 and the paper's applications, not a deep flaw; a revision that weakens condition (ii) to a large-λ requirement should resolve it. The independence worry in the reader's report about Case (Uni) is incorrect. I also recommend that the authors tighten the stochastic domination in Proposition 1.7 and include the decoupling inequality behind Lemma 2.4. The paper fits the journal's scope, and I saw no citation or novelty concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Straight to it: this paper does something real. It generalizes Hilário–Sá–Sanchis–Teixeira's percolation phase transition from randomly stretched lattices to oriented bond-site percolation with columnar disorder, under the same (1+ε)-moment condition, and then uses that to prove survival for contact processes with periodic recovery (both continuous and discrete random shifts) and for a Cox-environment contact process. The renormalization scheme is explicitly adapted from [12]—new stationary embedding for mixed bond-site, oriented crossing arguments, and the good/bad block definition tweaked—and the applications are new. The couplings for the contact processes are worked out in real detail. That is a substantial contribution.\n\nThe soft spot is real, but it is in the statement, not the core argument. Theorem 1.2 condition (ii) demands exp(-σs) ≤ κ_λ(s) for every λ≥0. For the paper's own κ_λ(s)=1-exp(-λe^{-s}) and for the block-counting κ_λ in the discrete case, κ_0(s)=0, so no σ>0 can work. The proof itself only establishes the bound for λ≥2 (Uni) or λ≥100 (Ber), then uses large λ to handle small s. That is all the renormalization actually needs: condition (ii) is only invoked for λ large, to make q_k(λ)→0 and to control the bad-block crossing probability with an exponential lower bound after scaling σ to 1. The local fix is to replace 'for every λ≥0' with 'for every λ≥λ0' for some λ0≥0, or to add a separate large-λ assumption. Without that amendment, the theorem is not satisfied by the paper's own examples.\n\nA reader's worry that the Case (Uni) coupling produces dependent stretches does not hold up. For i.i.d. uniforms U_i, twice the torus distance 2d_T(U_i,U_{i+1}) is independent of U_i (and of U_{i+1} by symmetry), and conditional on U_{i+1}, the two adjacent distances are independent uniforms. So the sequence (S_i) is i.i.d. uniform; ν_i=-log S_i is i.i.d. exponential. The environment is i.i.d. as required.\n\nThe rest of the proof looks sound: Lemma 2.4 is lifted from [12] with the moment condition used only there; the induction for crossing probabilities follows the standard lines; the Borel–Cantelli step is clean. The paper is honest about what it builds on and does not oversell.\n\nWho is this for: people working on stretched lattices, columnar disorder, and generalized contact processes. It deserves a serious referee; the theorem statement should be corrected, and no other load-bearing gaps are apparent. With that local fix I would take the result.","headline":"Genuine extension of the stretched-lattice phase transition to oriented bond-site percolation, with new contact-process applications; the theorem's condition (ii) is misstated for λ=0, but the fix is local and the paper deserves a serious referee.","tokens_in":18501,"tokens_out":8518,"would_cite":true,"duration_ms":84512,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K05","60K35","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"A $(1+\\varepsilon)$-moment condition on column stretches is enough to force percolation at large $\\lambda$, and the result implies survival of contact processes with periodic recovery.","keywords":["oriented percolation","random environment","columnar disorder","randomly stretched lattice","contact process","periodic recovery","generalised contact process","multiscale renormalisation"],"falsifier":"Compute the covariance of the consecutive stretches $\\nu_{i,i+1}=-\\log(2\\,d_T(U_i,U_{i+1}))$ in the continuous periodic-recovery case: because $U_i$ and $U_{i+1}$ are shared between neighbouring stretches, the covariance is nonzero, which shows the environment is not i.i.d.; alternatively evaluate $\\kappa_0(1)=0$ in that example, which violates the theorem's condition $\\exp(-\\sigma)\\le \\kappa_\\lambda(1)$ for every $\\lambda\\ge0$.","tokens_in":17439,"feed_emoji":"🦠","tokens_out":14434,"duration_ms":130609,"temperature":0.7,"pith_summary":"The paper establishes a percolation phase transition for oriented bond-site percolation on the planar lattice when entire spatial columns are randomly stretched: as long as the stretches have a finite $(1+\\varepsilon)$-th moment and the connection functions satisfy a mild exponential lower bound, there is a finite critical value $\\lambda_c$ such that percolation occurs for every $\\lambda>\\lambda_c$, almost surely in the environment. This extends earlier stretched-lattice results to a mixed bond-site setting with columnar disorder, where dependencies along columns do not decorrelate. The authors then couple this percolation model to generalised contact processes and prove survival for sufficiently large infection rates when recoveries are periodic with a random shift, and for small recovery rates in a random spatial environment with unbounded intensities. A phase transition follows because earlier extinction results cover small $\\lambda$. The paper also shows that temporal stretches with heavier-than-exponential tails remove the phase transition entirely.","feed_headline":"Stretched columns still percolate above a critical rate","feed_subtitle":"A (1+ε)-moment condition is enough, and it yields survival for contact processes with periodic recovery.","key_machinery":"The load-bearing object is the static multiscale renormalisation scheme: a stationarised renewal embedding of the environment with interarrival times $\\xi_i+\\nu_{i,i+1}$, a hierarchy of blocks whose sizes grow super-exponentially ($L_k=L_{k-1}\\lfloor L_{k-1}^{\\gamma-1}\\rfloor$), and a definition of good blocks that permits at most one bad sub-block (or two consecutive bad sub-blocks). Lemma 2.4, taken from the stretched-lattice literature, uses the $(1+\\varepsilon)$-moment condition through a decoupling inequality to bound the probability of a bad block by $L_k^{-\\alpha}$. On good rectangles, Lemmas 2.7 and 2.8 show that left-right, right-left, and bottom-top crossings fail with probability at most $\\exp(-L_k^\\beta)$; planarity then patches these crossings into an infinite path. The connection-function condition $\\exp(-\\sigma s)\\le \\kappa_\\lambda(s)$ is the one that lets even the longest stretches be crossed with probability bounded away from zero once $\\lambda$ is large.","core_discovery":"The central claim, Theorem 1.2, is that OPRE percolates under the sole moment condition $\\mathbb{E}[\\xi^{1+\\varepsilon}], \\mathbb{E}[\\nu^{1+\\varepsilon}]<\\infty$ and connection functions that are monotone, satisfy $\\exp(-\\sigma s)\\le \\kappa_\\lambda(s)$ with $\\kappa_\\lambda(s)\\to 1$ as $\\lambda\\to\\infty$: there exists $\\lambda_c<\\infty$ such that for every $\\lambda>\\lambda_c$ the model contains an infinite open path for almost every environment. The proof embeds the stretched columns stationarily as a renewal process, partitions space into scales of super-exponentially growing blocks, declares blocks good unless they contain too many bad sub-blocks, and bounds horizontal and vertical crossing failures by $\\exp(-L_k^\\beta)$. The applications are the paper's payoff: for the contact process with periodic recoveries of the form $2(\\mathbb{Z}+U)$ or $2\\mathbb{Z}+B$ on the line, survival occurs for all sufficiently large $\\lambda$; and for the contact process in a random environment with unbounded recovery intensities, survival is proved for small $\\delta$ under only a $(1+\\varepsilon)$-moment condition on the high intensity.","pith_inferences":["If the decoupling inequality can be proved under weaker dependence, the same scheme should apply to environments with short-range correlations between columns; a concrete test is to run the bad-block estimate on moving-average stretches.","The CPPR coupling suggests that survival should persist when periodic recoveries are replaced by almost-deterministic interarrival times with bounded support, since only the moments of the stretch variables enter the OPRE conditions.","The paper's time-limitation mechanism suggests a quantitative prediction for the Bernoulli-shift case: crossing a block of length $k$ costs about $\\lambda^k/k!$, so the critical infection rate should scale roughly linearly with $k$, which is testable by simulation.","The complementary periodic-infections setting, which the paper leaves open, is a natural candidate for the same coupling and may exhibit a non-trivial extinction phase in one dimension."],"forward_implications":["The contact process with periodic recovery in both the continuous shift and Bernoulli shift cases has a non-trivial phase transition: extinction for $\\lambda\\le(4d)^{-1}$ and survival for all sufficiently large $\\lambda$.","The contact process in a random spatial environment survives for every $\\delta$ below a positive critical value, even when the high recovery intensity has only a finite $(1+\\varepsilon)$-moment.","Any family of connection functions satisfying monotonicity and the exponential lower bound yields OPRE percolation for large $\\lambda$, so the theorem applies uniformly across vertex and edge environments.","For oriented percolation with temporal stretches of heavier-than-exponential tails, no phase transition exists: percolation fails for every $p\\in(0,1)$.","The $(1+\\varepsilon)$-moment assumption enters only through the decoupling inequality, leaving open whether finite first moments would suffice."],"supporting_citations":[{"why":"Supplies the multiscale renormalisation and the decoupling/bad-block estimates that Theorem 1.2 adapts to the oriented bond-site setting.","marker":"[12]"},{"why":"Provides the refined moment condition and is cited alongside [12] for the decoupling inequality behind Lemma 2.4.","marker":"[8]"},{"why":"Gives the extinction criterion for small recovery rates used to identify the lower phase in the CPPR phase transition.","marker":"[7]"},{"why":"Is the random-environment contact process result that Proposition 1.7 strengthens to unbounded recovery intensities.","marker":"[1]"},{"why":"Introduces the generalised contact process with random closed sets that Definition 1.4 extends.","marker":"[13]"},{"why":"Introduced the randomly stretched lattice that OPRE generalises to the oriented bond-site case.","marker":"[18]"},{"why":"Establishes survival for one-dimensional renewal contact processes with bounded support, the partial answer that the periodic-recovery results extend.","marker":"[23]"}],"fun_headline_variants":["Weak moment condition still yields percolation and survival","Stretched columns percolate under just a (1+ε) moment","Percolation and contact survival from a (1+ε)-moment bound","Column stretches: percolation and periodic recovery survive"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the random stretches in different columns are independent and have a finite $(1+\\varepsilon)$-th moment, yet in the periodic-recovery coupling consecutive stretches share a random uniform and are therefore dependent, and the required exponential lower bound is verified only for large $\\lambda$.","fun_headline_variants_meta":{"raw":{"variants":["Weak moment condition still yields percolation and survival","Stretched columns percolate under just a (1+ε) moment","Percolation and contact survival from a (1+ε)-moment bound","Column stretches: percolation and periodic recovery survive"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000195,"raw_usage":{"total_tokens":1307,"prompt_tokens":846,"completion_tokens":461,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":388}},"tokens_in":462,"tokens_out":461,"duration_ms":5989,"temperature":1.0,"reasoning_tokens":388,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:19:55.419345+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the covariance of the consecutive stretches $\\nu_{i,i+1}=-\\log(2\\,d_T(U_i,U_{i+1}))$ in the continuous periodic-recovery case: because $U_i$ and $U_{i+1}$ are shared between neighbouring stretches, the covariance is nonzero, which shows the environment is not i.i.d.; alternatively evaluate $\\kappa_0(1)=0$ in that example, which violates the theorem's condition $\\exp(-\\sigma)\\le \\kappa_\\lambda(1)$ for every $\\lambda\\ge0$.","supporting_citations":[{"cited_title":"Phase transition for percolation on a randomly stretched square lattice","cited_arxiv_id":null,"evidence_quote":"Supplies the multiscale renormalisation and the decoupling/bad-block estimates that Theorem 1.2 adapts to the oriented bond-site setting."},{"cited_title":"Phase transition on a randomly horizontally stretched square lattice","cited_arxiv_id":null,"evidence_quote":"Provides the refined moment condition and is cited alongside [12] for the decoupling inequality behind Lemma 2.4."},{"cited_title":"Contact process under renewals II","cited_arxiv_id":null,"evidence_quote":"Gives the extinction criterion for small recovery rates used to identify the lower phase in the CPPR phase transition."},{"cited_title":"Percolation in a dependent random environment","cited_arxiv_id":null,"evidence_quote":"Introduced the randomly stretched lattice that OPRE generalises to the oriented bond-site case."}],"review_version":1}