{"id":"8948b550-af09-4b4b-8573-87c76d25ab1f","arxiv_id":"2505.22471","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For sparse locally convergent graph sequences, the exponential fast/slow threshold is bounded below by the local limit's survival threshold, and stretched-exponential extinction is possible below it.","lead":"This paper proves general conditions under which the fast-extinction threshold of the contact process on large sparse random graphs matches the survival threshold of the infinite local limit. It also constructs sparse graphs where extinction is stretched exponential, showing that the polynomial time scale is not universal.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central inequality λ+ ≥ λ1 appears sound; the apparent sign issue in (13) is resolved by choosing ε and δ so that the exponential exponent is smaller than c, and the local typo in Prop. 1.7 is not used in the proof of Theorem 1.5.","rationale":"I read the strongest claim as Theorem 1.5, the general inequality λ+ ≥ λ1 for sparse locally converging graphs, which is also the step that resolves the second equality in Conjecture 1.2. I scrutinized the proof of λρ = λ+, the main engine of the theorem. The apparent sign error in the extinction-attempt bound is not an error: the probability of a successful extinction attempt is p = (1-e^{-1})^{εN} e^{-2λδN}, so the reciprocal is e^{c1εN+2λδN}, and multiplying by T^{-1} = e^{-cN} gives a vanishing term provided c1ε+2λδ < c. Since c is fixed and ε,δ can be decreased while preserving Lemma 2.5, the proof goes through. The argument for γn → 0 is also sound: on K ≤ T/3, the process spends at most 1/3 of [0,T] in low-density states because the low set is covered by disjoint unit intervals starting at the τk, so the uniform-time density is high at least 2/3 of the time, contradicting E[ρ(W)] → 0 obtained from λ < λρ and duality. I also checked that Theorem 1.4 is consistent with Theorem 1.5: its stretched-exponential lower bound is for a time scale much smaller than e^{cN}, so it only shows λ_- = 0, not a failure of λ+. The one genuine localized mistake I found is in the proof of Proposition 1.7's converse, where the displayed Markov chain bounds P(τ_R(o_n) ≤ t(n)) by a smaller event incorrectly. However, Markov applied directly to the event appearing in condition (4) proves the intended implication, and Proposition 1.7 is not used in Theorem 1.5. The over-broad statement of Corollary 1.6 (no moment assumptions on μ) is a statement-level hygiene issue; under the intended λ1(T) > 0 regime the degree distribution has exponential tail, so the theorem applies. Because the central inequality and its application to configuration models are supported, I do not think a load-bearing objection lands. The reader's conditional verdict remains reasonable due to the local proof issues, so I leave it unchanged.","tokens_in":17078,"tokens_out":41716,"duration_ms":499909,"concrete_test":"Recompute inequality (13) with explicit constants: for fixed c > 0, choose δ < c/(4λ), then a smaller ε satisfying -log(1-e^{-1})ε < c/4, and verify that Lemma 2.5 still gives the degree bound for that ε; if the final term indeed tends to 0, the sign concern is not load-bearing. As a second check, replace the invalid second inequality in Proposition 1.7's proof by the direct Markov bound P(∑_v 1{ξ^v_t=∅, τ_R(v)<t} > εN) ≤ (1/ε) P(ξ^{o_n}_t=∅, τ_R(o_n)<t(n)) and confirm that the displayed implication still follows.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I found no load-bearing flaw in the paper's central claim, Theorem 1.5. The proof of λρ = λ+ is the key step. The term in (13) is 3T^{-1} e^{-log(1-e^{-1})εN+2λδN} = 3e^{(c1ε+2λδ-c)N} with c1 = -log(1-e^{-1}) > 0. Since c > 0 is fixed, one may choose δ < c/(4λ) and then ε sufficiently small (while preserving the conclusion of Lemma 2.5, which is monotone in ε) so that c1ε+2λδ < c; the term then vanishes. The convergence of γn = P(K ≤ T/3) follows from λ < λρ via the uniform-time argument and duality, independently of the choice of ε. This resolves the reader's 'suspicious sign' concern without changing the conclusion. The genuine invalid inequality in the converse direction of Proposition 1.7 can be fixed by applying Markov directly to the sum of indicators {ξ^v_t=∅, τ_R(v)<t}, which is exactly controlled by condition (4); this is a local error and Proposition 1.7 is not used in the proof of Theorem 1.5. Theorem 1.4 does not contradict Theorem 1.5 because its lower bound concerns the stretched-exponential scale exp(N/log^{1+ε}N), which is smaller than any fixed exponential scale e^{cN}; hence it says nothing about λ+. Corollary 1.6 suppresses moment assumptions, but in the regime λ1(T) > 0 the degree distribution has an exponential tail, so the sparse and local-convergence hypotheses are satisfied. For these reasons, the central claim survives scrutiny.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the contact process on sequences of finite sparse random graphs that converge locally in probability. It introduces a metastable density and shows (Theorem 1.3) that the asymptotic infection density at any diverging time is bounded above by the annealed survival probability of the local limit. It constructs a sparse locally converging sequence for which the extinction time is stretched exponential for every infection rate (Theorem 1.4), so fast extinction on the polynomial scale fails. The main result (Theorem 1.5) states that for sparse locally converging sequences, the fast/slow threshold λ+ is at least the survival/extinction threshold λ1 of the limit graph; combined with the upper bound of Nam–Nguyen–Sly, this yields λ+ = λ1 for configuration-model giant components (Corollary 1.6). Proposition 1.7 characterizes when the metastable density converges to the limit's survival probability, and Lemma 1.9 links tightness of extinction times to subcriticality in the limit.","tokens_in":17400,"tokens_out":33393,"duration_ms":332361,"significance":"If correct, Theorem 1.5 gives a general locality principle for the phase transition of the contact process and confirms the second equality in Conjecture 1.2. The metastable-density viewpoint is a useful addition to the subject, and the proof of the lower bound λ+ ≥ λ1 is essentially self-contained, relying on external work only to identify local limits and to supply the complementary upper bound. The construction in Theorem 1.4 clarifies that the exponential scale is the natural separation scale in sparse graphs. These contributions justify publication in a serious probability journal.","major_comments":[],"minor_comments":[{"comment":"The displayed inequality in the second half of the proof replaces the sum of indicators {ξ_v_t(n)=∅, τ_R^(n)(v)<t(n)} by #{v: τ_R^(n)(v) ≤ t(n)}, but τ_R(v) ≤ t(n) does not imply ξ_v_t(n)=∅, so the intermediate bound is invalid. The intended conclusion follows directly by applying Markov's inequality to the original sum, which yields (1/ε) P(ξ_{o_n}_t(n)=∅, τ_R^(n)(o_n)<t(n)). This is a local gap and does not affect the proof of Theorem 1.5.","section":"Section 2.4, proof of Proposition 1.7"},{"comment":"The exponent in the last term of (13) is correct once ε and δ are chosen so that −log(1−e^{−1}) ε + 2λδ < c, but the sentence 'Decreasing the values of ε and δ if needed' should explicitly note that Lemma 2.5 allows ε to be taken arbitrarily small as δ → 0, so the required inequality can be achieved for each fixed c > 0.","section":"Section 2.5, Eq. (13)"},{"comment":"The proof does not explicitly verify that the constructed sequence of augmented Gilbert graphs is sparse in the sense of uniform integrability of the degree of a uniformly chosen vertex. This follows from the finite mean of the radius distribution and the local convergence, but the verification should be stated.","section":"Theorem 1.4 and its proof"},{"comment":"The claim that on the event {K ≤ T/3} the proportion of time spent in high-density states satisfies 1 − r(ε,T) ≥ 2/3 is not justified in the text. A short argument using that each low-density interval of length ℓ contains at most ℓ + O(1) of the stopping times τ_k gives the bound, but the paper should include it or cite a lemma.","section":"Section 2.5, paragraph after Eq. (13)"},{"comment":"There are several typographical issues: 'distrbution' in Theorem 1.5, '/emptysetstress' and '/BD' artifacts from the LaTeX source, and 'procoess' in Section 2.1. These should be corrected in revision.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The paper's central claim, Theorem 1.5, appears sound; the only proof error is in Proposition 1.7, which is not used in the proof of the main theorem and is easily corrected. The self-citation [21] is used only to identify the local limit of the constructed example, which is appropriate. The manuscript fits the scope of math.PR and is likely to be of interest to researchers in interacting particle systems and random graphs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is a legitimate step forward. It proves λ+ ≥ λ1 for sparse locally converging graph sequences with extremal local limit, and together with the known upper bound from Nam–Nguyen–Sly this resolves the second equality in their conjecture for configuration models. That is a real result, not a repackaging of existing techniques. The metastable-density perspective (Theorem 1.3 and Proposition 1.7) is a clean way to formulate the fast/slow dichotomy, and the stretched-exponential example in Theorem 1.4 is a nice counterpoint to purely polynomial extinction regimes.\n\nI went through the proof of Theorem 1.5 carefully. The central step, proving λρ = λ+, works. The term in (13) that looked like a sign error is actually 1/p times T^{-1} with p the per-attempt extinction probability; its exponent is c1ε + 2λδ − c, and the text's instruction to decrease ε and δ is legitimate. So that concern dissolves.\n\nThere is a genuine local error in the proof of Proposition 1.7. In the second implication, the paper bounds the count of vertices with ξv_t=∅ and τR(v)<∞ by the count of all vertices with τR(v)≤t, then replaces P(τR(o)<t) by P(ξo_t=∅, τR(o)<t) in the wrong direction. The fix is immediate: apply Markov directly to the original sum, whose expectation is exactly |Vn| P(ξo_t=∅, τR(o)<t). Proposition 1.7 is not used in the proof of Theorem 1.5, so the main line survives.\n\nTwo smaller points. Theorem 1.4 relies on the companion paper [21] for the identification of the local limit and its positive survival threshold; that is fine, but the dependence is worth making explicit. And Corollary 1.6 suppresses moment conditions; in the regime λ1(T)>0 the degree distribution has exponential tail, so the sparsity and local-convergence hypotheses are met, but a reader not familiar with [7,20] will have to do some work.\n\nBottom line: the paper deserves a serious referee. The central inequality is new and the proof strategy is sound. The Proposition 1.7 proof needs a corrected inequality chain, and the exposition around Theorem 1.4 could be more self-contained. I would accept it with minor revisions.\n\nBest","headline":"A genuinely new and mostly sound proof of the λ+ ≥ λ1 inequality for sparse locally converging graphs, resolving the second equality of the Nam–Nguyen–Sly conjecture for configuration models; the one real proof error is local and easily fixed.","tokens_in":17989,"tokens_out":8296,"would_cite":true,"duration_ms":82538,"reading_group":"yes","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":"For sparse random graphs, the fast/slow infection threshold is governed by the local limit.","keywords":["contact process","metastability","fast/slow extinction threshold","local weak convergence","sparse random graphs","configuration model","Galton-Watson tree","survival threshold"],"falsifier":"Run the contact process from full occupancy on a configuration-model giant component with an exponentially-tailed degree distribution, at a rate $\\lambda$ strictly below $\\lambda_1(T)$. If for some $c>0$ the probability of surviving $e^{c|V_n|}$ does not decay to $0$, then $\\lambda_+<\\lambda_1(T)$, contradicting Corollary 1.6; the theorem predicts subexponential extinction for every such $\\lambda$.","tokens_in":16808,"feed_emoji":"🦠","tokens_out":6551,"duration_ms":61483,"temperature":0.7,"pith_summary":"This paper studies when the contact process on a growing sequence of finite sparse random graphs dies out quickly versus survives exponentially long. The authors propose metastability as the organising principle: the density of infected vertices at large times can be compared to the survival probability of the contact process on the infinite local limit. Their main theorem states that, for sparse connected graphs converging locally in probability to an extremal random graph, the fast/slow threshold is at least the survival threshold of the limit; combined with an earlier upper bound, this proves the second equality in a conjecture of Nam, Nguyen and Sly for configuration-model giant components. They also construct sparse scale-free spatial graphs where extinction, while fast relative to the exponential scale, occurs on stretched exponential time scales for every infection rate, showing that exponential is the natural separating scale in general.","feed_headline":"Sparse networks: fast/slow infection threshold set by limit tree","feed_subtitle":"For configuration-model giant components, the fast/slow threshold equals the survival threshold of the limiting Galton-Watson tree.","key_machinery":"The key object is the metastable density $\\eta_\\lambda(Q) = \\mathbb{E}[P^\\lambda_G(\\tau(o)=\\infty)]$, the annealed survival probability of the contact process on the local limit, together with the density process $\\rho(t)=|\\xi_t|/|V_n|$. Theorems 1.3 and Proposition 1.7 show that, under local convergence in probability to an extremal limit, $\\rho(t(n))$ converges in probability to $\\eta_\\lambda(Q)$ exactly when infection does not escape the $R$-neighbourhood of a typical root before time $t(n)$, as encoded in condition (4). The sparsity condition, uniform integrability of the degrees, enters through Lemma 2.5, which bounds the total degree of any small infected set; this lets the proof force extinction at each low-density visit of the process. Theorem 1.4 uses survival on star graphs, fed by a vertex of maximal degree in an augmented Gilbert graph with power-law radii.","core_discovery":"The central claim is Theorem 1.5: if $(G_n)$ is a sequence of connected sparse random graphs converging locally in probability to an extremal random graph $(G,o)$, then $\\lambda_+((G_n)) \\ge \\lambda_1(G)$, where $\\lambda_+$ is the fast/slow extinction threshold of the finite graphs and $\\lambda_1$ is the survival/extinction threshold of the infinite limit. With the upper bound from Nam, Nguyen and Sly, this yields $\\lambda_+((C_n)) = \\lambda_1(T)$ for the giant components of configuration models, where $T$ is the associated unimodular Galton-Watson tree, confirming the second equality of Conjecture 1.2. The proof works by showing that above $\\lambda_1$ the infection density cannot vanish on exponential time scales: absence of metastability on the exponential scale forces extinction, and sparse graphs cannot support survival beyond exponential scales.","pith_inferences":["The metastable-density mechanism should transfer to other monotone interacting particle systems on locally converging sparse graphs, whenever the limiting object has a well-defined survival probability that plays the role of $\\eta_\\lambda(Q)$.","The paper's star-fed construction suggests that a hierarchy of stretched-exponential time scales can be produced by varying the tail of the radius distribution; the authors note the lower bound is not optimal, so one can ask whether $\\exp(\\Theta(|V_n|))$ is reachable for subcritical rates in some sparse graphs.","The role of extremality deserves separate scrutiny: without an extremal limit, annealed and quenched survival probabilities may differ, and the equality $\\lambda_+ = \\lambda_1$ may fail even under sparsity."],"forward_implications":["For configuration-model giant components with degree distribution satisfying $\\sum_k k(k-2)\\mu(k)>0$, the fast/slow threshold equals the survival threshold of the associated unimodular Galton-Watson tree, confirming the second equality of Conjecture 1.2.","For every sparse connected graph sequence converging locally in probability to an extremal limit, there is no fast extinction below the limit's survival threshold: the finite-graph threshold is at least $\\lambda_1(G)$.","The exponential scale is a canonical separator: sparse graphs cannot exhibit survival on super-exponential time scales, and the constructed scale-free spatial graphs show that extinction slow enough to be called 'fast' can still occur on stretched exponential scales for every $\\lambda>0$.","If condition (4) holds, the infection density at diverging times converges in probability to the annealed survival probability of the local limit, a law of large numbers for the metastable density.","Under sparsity, the density-based threshold $\\lambda_\\rho$ coincides with the extinction-time threshold $\\lambda_+$, so the two definitions of the fast/slow boundary are equivalent."],"supporting_citations":[{"why":"supplies the upper bound $\\lambda_+((C_n))\\le \\lambda_1(T)$ for configuration-model giant components, completing the equality with Theorem 1.5.","marker":"[28]"},{"why":"establishes local convergence in probability of giant components to the unimodular Galton-Watson tree, the setting in which Corollary 1.6 applies.","marker":"[18]"},{"why":"defines and analyses the augmented Boolean model that is the local limit in Theorem 1.4, proving its survival threshold is positive for finite-mean radii.","marker":"[21]"},{"why":"provides the initial exponential survival result on star graphs used to lower-bound extinction times in the scale-free spatial construction.","marker":"[4]"},{"why":"gives the exponential lower bound on the expected extinction time of a star and the universal supercritical extinction-scale bound that the construction is matched against.","marker":"[30]"},{"why":"supplies the tail estimate $P(\\tau\\le t)\\le t/\\mathbb{E}[\\tau]$ on stars, used to turn the expectation bound into a high-probability survival statement.","marker":"[31]"},{"why":"shows that without sparsity the contact process can survive on super-exponential time scales, delimiting the scope of the exponential-scale picture.","marker":"[8]"}],"fun_headline_variants":["Phase boundary on sparse graphs set by limit tree","Fast-slow extinction threshold equals survival threshold on sparse graphs","Metastability reveals phase transition in sparse random graphs","Limit tree sets infection threshold on sparse graph contact process","Sparse graph phase boundary: limit tree decides fast/slow extinction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on sparsity, meaning the vertex degrees are uniformly integrable under the contact-process law; if that fails, small infected sets can carry large total degree, Lemma 2.5 collapses, and the paper itself cites examples with super-exponential survival.","fun_headline_variants_meta":{"raw":{"variants":["Phase boundary on sparse graphs set by limit tree","Fast-slow extinction threshold equals survival threshold on sparse graphs","Metastability reveals phase transition in sparse random graphs","Limit tree sets infection threshold on sparse graph contact process","Sparse graph phase boundary: limit tree decides fast/slow extinction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000514,"raw_usage":{"total_tokens":2478,"prompt_tokens":910,"completion_tokens":1568,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":1489}},"tokens_in":526,"tokens_out":1568,"duration_ms":11598,"temperature":1.0,"reasoning_tokens":1489,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:06:50.830852+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the contact process from full occupancy on a configuration-model giant component with an exponentially-tailed degree distribution, at a rate $\\lambda$ strictly below $\\lambda_1(T)$. If for some $c>0$ the probability of surviving $e^{c|V_n|}$ does not decay to $0$, then $\\lambda_+<\\lambda_1(T)$, contradicting Corollary 1.6; the theorem predicts subexponential extinction for every such $\\lambda$.","supporting_citations":[{"cited_title":"Critical value asymptotic s for the contact process on random graphs","cited_arxiv_id":null,"evidence_quote":"supplies the upper bound $\\lambda_+((C_n))\\le \\lambda_1(T)$ for configuration-model giant components, completing the equality with Theorem 1.5."},{"cited_title":"On the spr ead of viruses on the internet","cited_arxiv_id":null,"evidence_quote":"provides the initial exponential survival result on star graphs used to lower-bound extinction times in the scale-free spatial construction."},{"cited_title":"The contact process on random graphs","cited_arxiv_id":null,"evidence_quote":"supplies the tail estimate $P(\\tau\\le t)\\le t/\\mathbb{E}[\\tau]$ on stars, used to turn the expectation bound into a high-probability survival statement."},{"cited_title":"Super-exponential extinction time of the contact process on random geometric graphs","cited_arxiv_id":null,"evidence_quote":"shows that without sparsity the contact process can survive on super-exponential time scales, delimiting the scope of the exponential-scale picture."}],"review_version":1}