{"id":"812a3523-bc89-4199-80dd-3ecb4c55e3c8","arxiv_id":"2501.16858","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Contact processes on stationary one-dimensional random graphs have a non-trivial survival/extinction phase transition whenever the number of long-range edges crossing each link is finite.","lead":"This paper proves that a simple epidemic model on certain one-dimensional random networks has a genuine slow-spread regime where the disease always dies out. It settles a 2015 conjecture for long-range connections and shows the same holds even when vertex degrees are heavy-tailed.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 2.2 asserts that the block-index process Z is a nearest-neighbor random walk in a stationary ergodic environment, but the transition probabilities of Z as defined are not shown to depend only on the block environment; the Ledrappier step needs an explicit regeneration or…","rationale":"The theorem's claim λ_c>0 under (A.1), (A.2), and P(e<∞)=1 is plausible and the surrounding applications are valuable, but the proof's pivotal reduction to Ledrappier's criterion is currently asserted rather than demonstrated. The reader's weakest_assumption focused on P(e<∞)=1; that is indeed a real limitation, but it is an explicitly stated hypothesis, not an internal gap. The Markovianity of Z is more load-bearing because without it Proposition 3.4 cannot be invoked at all. I do not recommend rejection: the gap may be fillable by a standard regeneration or domination argument, and the rest of the argument, including finite expected block size and edge count and the positivity of E log(1/ω), is consistent. However, the verdict should stay CONDITIONAL until the construction is supplied, because the current text does not prove the central theorem as written.","tokens_in":19990,"tokens_out":31533,"duration_ms":306883,"concrete_test":"Carry out the following check on Proposition 3.6: define Z̃ as the block chain observed at successive visits to the left boundary of a block, with all vertices to the left permanently infected in the dominating process, and prove (i) its quenched transition probabilities from block k are functions of (C_k,E(C_k)) only; (ii) these probabilities form a stationary ergodic sequence under the Palm measure; (iii) recurrence of Z̃ implies Pλ(∩_{n≥1}∪_{t>n}{η_t={0}})=1. If (i) fails, exhibit a block with an internal clique and two entry configurations compatible with the reset rule for which the probability of exiting to the right differs, disproving the paper's Markovianity assertion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3.1, after defining the dominating process η, the maximum X_t, the block index Y_t, and the jump chain Z_n of Y, the text states that 'Z is a random walk in a stationary ergodic random environment on N∪{0}' and then applies Proposition 3.4 (Ledrappier). This is the step on which Theorem 2.2 rests. However, as defined, Z records the block of the right-most infected vertex at every time the block index changes. The probability that the next change is to block k+1 rather than k−1 depends not only on the graph block C_k but on the full infection configuration inside C_k at the current time, including which vertices have already recovered and how long ago the left half-line was refilled. Attractiveness provides inequalities of the form (12), but it does not make the jump probabilities a function of the environment alone; a continuous-time Markov process projected to a coarse block index is generally not Markov, let alone a nearest-neighbor RWRE. The K>1 case adds the further issue that, even with (K,L)-cut points, several edges can cross a boundary, so the exit probability depends on which left endpoints are infected. Thus the hypothesis of Ledrappier's theorem is not verified for the process Z as constructed. The proof needs a regeneration construction: for example, observe the block chain only at successive visits to the left boundary of a block, or use a dominating process with a permanently infected left half-line, and prove that the resulting transition probabilities are stationary, ergodic, and block-dependent. Without this, the central recurrence argument is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the contact process on stationary ergodic random graphs with vertex set Z, augmented by long-range edges. The main result, Theorem 2.2, states that if the random graph satisfies (A.1) stationarity and ergodicity, (A.2) finite expected root degree, and the number e of edges crossing a single link is almost surely finite, then the critical infection rate satisfies λ_c∈(0,∞) almost surely. The proof decomposes the graph into blocks between cut points, constructs a dominating contact process whose rightmost infection is tracked, and claims that the block-index jump chain is a random walk in a stationary ergodic random environment; Ledrappier's criterion is then invoked to show recurrence for small λ. Applications are given to long-range percolation (improving Can's bound to δ>2), augmented Gilbert graphs, weight-dependent random connection models, and a higher-dimensional no-phase-transition result for heavy-tailed Boolean models.","tokens_in":20336,"tokens_out":45201,"duration_ms":391864,"significance":"If correct, Theorem 2.2 is a broad and simple sufficient condition for nontrivial phase transitions on one-dimensional spatial random graphs, covering heavy-tailed degree distributions where earlier methods required exponential tails. The paper is generally well written, and the applications to long-range percolation (Can's conjecture for δ>2) and to sharp Gilbert-graph results are attractive. The proof idea of coupling the block process to a random walk in random environment is novel and potentially useful. However, the central RWRE coupling is not established as written, and the proof of one of the advertised applications contains a false inference.","major_comments":[{"comment":"The process Z, defined as the block index at jump times of Y, is not a Markov chain whose transition probabilities depend only on the block C_k. At the start of a sojourn in block k, the rightmost infected vertex is at the left boundary z_{k-1} if the previous block transition was from k-1 to k, but at the right boundary z_k-1 if it was from k+1 to k; the probabilities of the next exit being to k+1 versus k-1 differ in the two cases (for a two-vertex block with nearest-neighbour edges and λ<1, the right-exit probability from the left boundary is λ^2/((λ+1)^2-λ), while from the right boundary it is strictly larger). Hence the 'crucial observation' that Z is a random walk in a stationary ergodic environment is not established, and Ledrappier's Proposition 3.4 cannot be applied to the process as constructed. Moreover, since 0 is permanently infected in ξ†, the first block C_1 has no left exit, so the right-jump probability ω_1(λ) equals 1, contradicting the claim that ω_1(λ)↓0 as λ↓0. A regeneration construction (e.g., sampling the chain only at successive crossings of a fixed cut point, or enlarging the state to include the entry direction) is needed.","section":"Section 3.1"},{"comment":"The generator written there has ω_x as the coefficient of f(x-1), i.e., ω_x is the left-jump probability, but the recurrence criterion ∫ log((1-ω_1)/ω_1) dQ ≥ 0 is the criterion for ω_x being the right-jump probability. For the generator as stated, the correct criterion is ∫ log(ω_1/(1-ω_1)) dQ ≥ 0. Since the proof of Proposition 3.6 defines ω_k(λ) as the right-jump probability Pλ(Z_{k+1}=z+1|Z_k=z), the proposition as stated is inconsistent with its use and must be corrected.","section":"Proposition 3.4"},{"comment":"The paper claims that if the dominating process η returns to state {0} infinitely often, then the original process ξ^{{-1,0}} dies out. This implication is not immediate and is not proved: survival of ξ^{{-1,0}} is compatible with ξ†_t={0} infinitely often, since at those times the original process could itself be at {0}. A regenerative argument showing that each excursion of the original process away from {0} has a uniformly positive probability of dying out is required; as written, the deduction of extinction from recurrence of Z is incomplete.","section":"Section 3.1, Eq. (11)"},{"comment":"The step 'the probability that 0 is not covered by any ball of the Boolean model is positive if and only if ∫ rρ(dr)<∞; hence ... 0 has a positive probability of being a cut point' is not valid. A gap in the Boolean model around the origin does not prevent a long edge from crossing the link ℓ_0: two balls centered on opposite sides of the origin can overlap across it without either covering the origin. Moreover, for a deterministic lattice with constant radius 1, ∫ r dρ<∞ but every link has e(z)≥3, so 1-cut points do not exist. The application should verify the hypotheses of Theorem 2.2 directly, e.g., by showing P(e<∞)=1, rather than relying on the uncovered-origin event.","section":"Proof of Theorem 2.5"}],"minor_comments":[{"comment":"The reference list contains a duplicate entry: [27] and [28] are both Huang and Durrett, 'The contact process on random graphs and Galton Watson trees', ALEA 17 (2020).","section":"References"},{"comment":"In Theorem 2.7, the displayed double integral in (7) is garbled: the lower limits of the two one-dimensional integrals should be 2^{-n-\\mu n}; as typeset, the formula is not readable.","section":"Theorem 2.7"},{"comment":"The phrase 'spatially at most two dependent' is imprecise; the oriented edges are spatially dependent with bounded range and bounded degree, which is sufficient for the Liggett-Schonmann-Stacey theorem, but the number of dependent edges is not two in general.","section":"Section 3.3"},{"comment":"Proposition 3.1 refers to the sequence (C_k)_{k≥0}, but the blocks C_k are defined in Section 3.1 only for k≥1; please clarify the definition of C_0 or index the statement consistently.","section":"Proposition 3.1"}],"recommendation":"major_revision","confidential_remarks":"The main result is potentially important and the paper is readable, but the central RWRE coupling in the proof of Theorem 2.2 is not currently justified. The issues are substantial yet seem local in the sense that a careful regeneration construction might repair the proof. The proof of Theorem 2.5 also contains a false assertion about the relation between uncovered origin and cut points. I recommend major revision, with a careful rewrite of Section 3.1 and of the verification of the Gilbert-graph application."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper has a genuinely new mechanism and strong results—it settles Can's δ*≤2, gives the first heavy-tailed spatial examples with a subcritical phase, and the sharp Gilbert-graph criterion is nice. But the proof of the central theorem has a gap that needs a real fix, not just wording.\n\nWhat's new: the cut-point plus random-walk-in-random-environment coupling replaces the cumulative merging percolation machinery. The applications are non-trivial: the WDRCM condition (7) is checked in detail, and the sharpness of Theorem 2.6 is a good addition. The paper is honest about the e=∞ obstruction (Remark 2.3) and about d>1 being conjectural.\n\nThe load-bearing problem is in Section 3.1. The block-index process Z is asserted to be a nearest-neighbor RWRE with transition probabilities ω_k that depend only on C_k. I don't think that's justified as written. Z changes only when X crosses a cut point, and at an upward crossing the state is canonical (all left infected, leftmost vertex of the new block infected). But at a downward crossing, the state is an all-infected interval ending at some random vertex b inside the lower block. The probability that the next Z-change is up depends on b (and, through the cut edge, on the previous block's structure), so the transition probabilities are not a function of the environment alone. Attractivity gives inequalities like (12), but it doesn't make Z Markov. The Ledrappier step needs a regeneration construction, e.g., observing the chain at successive visits to the left boundary of a block, or a genuinely block-dependent environment with a proof. Without that, Theorem 2.2 is unsupported.\n\nMinor issues: Prop 3.4's generator swaps the left/right jump coefficients relative to ω_k; the (K,L)-cut point block decomposition is sketched rather than proved; Theorem 2.8's renormalization is an outline. These are minor compared to the RWRE gap.\n\nI don't think the paper is circular: Ledrappier, Kac's lemma, and [22] are external and don't contain the target result. The writing is clear and the limitations are stated. If the RWRE step can be fixed, this is a strong paper. As it stands, I wouldn't rely on Theorem 2.2, but I would want to see a revision.\n\nRecommendation: send to a serious referee. The referee should ask for a rigorous construction of the block process and the RWRE coupling, and for a correction of Prop 3.4. The results justify the effort.","headline":"Strong results and a promising new mechanism, but the central RWRE step in Theorem 2.2 has a real gap that needs fixing before the proof holds.","tokens_in":20842,"tokens_out":23878,"would_cite":false,"duration_ms":196277,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","05C82","91D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A finiteness condition on edge crossings across each cut forces a positive epidemic threshold on stationary one-dimensional random graphs.","keywords":["contact process","phase transition","long-range percolation","Gilbert graph","random geometric graph","scale-free network","SIS epidemics","random walk in random environment"],"falsifier":"To test the theorem, simulate the contact process on an augmented Gilbert graph on $\\mathbb{Z}$ built from i.i.d. radii with finite mean, at a small infection rate such as $\\lambda=0.1$, on finite windows of growing length; the theorem predicts the infection dies out. If the survival probability stays bounded away from zero as the window grows, the asserted positive critical rate is false. A complementary test is to construct a stationary, ergodic, sparse graph on $\\mathbb{Z}$ with almost surely finite edge crossings per link and measure whether $\\lambda_c$ is positive; the theorem says every such graph has $\\lambda_c>0$.","tokens_in":19825,"feed_emoji":"🦠","tokens_out":11523,"duration_ms":101976,"temperature":0.7,"pith_summary":"The contact process is a basic model of infection spread: infected vertices pass the infection to neighbours at rate $\\lambda$ and recover at rate $1$. This paper asks when a random infinite network built on the integers has a genuine extinction phase, meaning the infection dies out for small $\\lambda$ and can survive only for large $\\lambda$. The main theorem says this happens whenever the network is stationary and ergodic, has finite expected degree, and almost surely only finitely many long edges cross each cut between the left and right half-lines. That condition is mild enough to cover heavy-tailed spatial networks where earlier techniques, which required exponentially decaying degrees, failed. In particular, the result settles the long-range percolation threshold $\\delta>2$ and gives a sharp integrability condition for geographic Gilbert graphs.","feed_headline":"Epidemics on sparse line graphs keep a subcritical phase","feed_subtitle":"If each cut is crossed by finitely many long edges, the infection dies out at low rates—even with heavy-tailed degrees.","key_machinery":"The central object is the cut-point block decomposition of $\\mathbb{Z}$. A link is a $K$-cut point when at most $K$ edges of the graph connect its left side to its right side; the almost-sure finiteness of $e$ makes cut points occur with positive density, and ergodicity makes the blocks between consecutive cut points a stationary sequence of finite random graphs with finite expected length and finite expected edge count. The proof then couples the contact process on these blocks to a nearest-neighbour random walk in a random environment on $\\mathbb{N}\\cup\\{0\\}$, where the environment is the random block structure. The key identity is the recurrence criterion $\\int \\log((1-\\omega)/\\omega)\\,dQ \\ge 0$ for the random walk's jump probabilities $\\omega$; when $\\lambda$ is small the block environment makes this integral positive, so the walk recurs, and recurrence of the block process is what converts survival into extinction.","core_discovery":"The paper's main claim is Theorem 2.2: for a random graph on $\\mathbb{Z}$ satisfying stationarity and ergodicity, finite expected root degree, and $\\mathbb{P}(e<\\infty)=1$, where $e$ is the number of edges crossing the link between $-1$ and $0$, the critical infection rate satisfies $\\lambda_c(G)\\in(0,\\infty)$ almost surely. The condition $\\mathbb{P}(e<\\infty)=1$ means cut points exist with positive density: links crossed by only one or a few edges split the line into finite blocks. The proof shows that the rightmost infected particle, viewed on the block chain, is controlled by a random walk in a random environment, and a classical recurrence criterion makes that walk recurrent for all sufficiently small $\\lambda$. Recurrence forces the infection to return to a single point infinitely often, which is exactly extinction. The same mechanism yields a subcritical phase for long-range percolation with connection probabilities of order $|x-y|^{-\\delta}$ for every $\\delta>2$, for augmented Gilbert graphs if and only if the radius distribution has finite mean, and it shows that in dimensions $d\\ge 2$ heavy power-law tails with exponent $\\tau\\le d+1$ destroy the subcritical phase.","pith_inferences":["Editorial extension: the recurrence criterion in the proof is quantitative, so the same block decomposition can yield explicit lower bounds for $\\lambda_c$ from the laws of block length and block edge count, which the paper does not work out.","Editorial extension: the theorem does not decide what happens when $e=\\infty$ almost surely; we infer that some of those graphs may still have $\\lambda_c>0$, and the critical scale-invariant long-range percolation model is a natural place to test this numerically.","Editorial extension: viewed as an epidemic statement, the result suggests that on one-dimensional heavy-tailed networks the way to create a true epidemic threshold is to suppress very long edges rather than to reduce mean degree; this policy reading is not in the paper."],"forward_implications":["Long-range percolation on $\\mathbb{Z}$ with connection probability of order $|x-y|^{-\\delta}$ has a positive epidemic threshold for every $\\delta>2$, verifying the standing conjecture and improving the older bound $\\delta^{*}\\le 102$.","For augmented Gilbert graphs on the line with i.i.d. radii, the subcritical phase exists if and only if the typical radius has finite mean; infinite mean makes the graph fail to be locally finite and forces $\\lambda_c=0$.","In dimensions $d\\ge 2$, spatial Boolean models with power-law degree exponent $\\tau\\le d+1$ have $\\lambda_c=0$, so without at least a finite $d$-th degree moment there is no extinction phase.","Heavy-tailed degree distributions do not by themselves rule out a non-trivial epidemic phase when long edges are geometrically scarce."],"supporting_citations":[{"why":"provides the recurrence criterion for random walks in random environment that proves the block process recurrent for small $\\lambda$.","marker":"[36]"},{"why":"gives the return-time relation used to show consecutive cut-point blocks have finite expected length.","marker":"[31]"},{"why":"supplies the one-dimensional long-range percolation condition that the finite-crossing assumption generalises.","marker":"[50]"},{"why":"formulates the long-range percolation conjecture and previous upper bound that Theorem 2.4 verifies.","marker":"[8]"},{"why":"provides the contrasting Galton–Watson-tree result showing exponential degree tails are needed without spatial structure.","marker":"[4]"},{"why":"supplies the continuum-percolation covering facts used to identify cut points and infinite degree in augmented Gilbert graphs.","marker":"[45]"},{"why":"gives the summability criterion for long edges used to verify cut points and sparsity for weight-dependent random connection models.","marker":"[22]"},{"why":"provides the star-graph survival estimates used in the higher-dimensional block renormalisation argument.","marker":"[28]"}],"fun_headline_variants":["Epidemics on sparse networks always have a subcritical phase","Subcritical phase proven for contact processes on random line graphs","Cut points guarantee extinction for low infection rates","Non-trivial phase transition on networks with heavy-tailed degrees","Sharp condition for epidemic survival on augmented lines"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire argument depends on almost every link of the integer line being crossed by only finitely many long edges; if infinitely many long edges cross every link almost surely, cut points have probability zero, the block decomposition never starts, and the random-walk coupling cannot be built.","fun_headline_variants_meta":{"raw":{"variants":["Epidemics on sparse networks always have a subcritical phase","Subcritical phase proven for contact processes on random line graphs","Cut points guarantee extinction for low infection rates","Non-trivial phase transition on networks with heavy-tailed degrees","Sharp condition for epidemic survival on augmented lines"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001026,"raw_usage":{"total_tokens":4443,"prompt_tokens":1181,"completion_tokens":3262,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":797,"completion_tokens_details":{"reasoning_tokens":3186}},"tokens_in":797,"tokens_out":3262,"duration_ms":19478,"temperature":1.0,"reasoning_tokens":3186,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T10:08:35.573040+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test the theorem, simulate the contact process on an augmented Gilbert graph on $\\mathbb{Z}$ built from i.i.d. radii with finite mean, at a small infection rate such as $\\lambda=0.1$, on finite windows of growing length; the theorem predicts the infection dies out. If the survival probability stays bounded away from zero as the window grows, the asserted positive critical rate is false. A complementary test is to construct a stationary, ergodic, sparse graph on $\\mathbb{Z}$ with almost surely finite edge crossings per link and measure whether $\\lambda_c$ is positive; the theorem says every such graph has $\\lambda_c>0$.","supporting_citations":[{"cited_title":"Quelques proprietes des exposants caracteristiques","cited_arxiv_id":null,"evidence_quote":"provides the recurrence criterion for random walks in random environment that proves the block process recurrent for small $\\lambda$."},{"cited_title":"On the notion of recurrence in discrete stochastic processes","cited_arxiv_id":null,"evidence_quote":"gives the return-time relation used to show consecutive cut-point blocks have finite expected length."},{"cited_title":"Contact process on one-dimensional long range percolation","cited_arxiv_id":null,"evidence_quote":"formulates the long-range percolation conjecture and previous upper bound that Theorem 2.4 verifies."}],"review_version":1}