{"id":"b0ec42e5-04ee-4d6c-b426-25eb1505068a","arxiv_id":"2502.01424","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The frozen Erdős-Rényi graph has a fluid limit governed by explicit ODEs, and its total gelation time rescaled by n converges to a Gumbel distribution with explicit constants.","lead":"This paper proves that the frozen Erdős-Rényi random graph, where components with a unique cycle stop growing, has a deterministic large-scale limit described by explicit differential equations. It also gives the exact random fluctuation of the time when the whole graph freezes.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The all-p free forest property (Prop. 1.3) is cited to an unpublished thesis; every jump and drift estimate in the proof of Theorem 1.2 depends on it, so a small-n exact enumeration is the decisive check.","rationale":"The reader's weakest-assumption analysis correctly identifies Proposition 1.3 as the most load-bearing underexposed step. The entire proof architecture for Theorem 1.2 — Corollary 2.5, Lemma 4.4, Lemma 4.9 — sits on the free forest property. The property is plausible and is known for p=1/2, but the all-p version is cited to an unpublished thesis, so the paper is not fully self-contained. I do not see an internal contradiction or a counterexample; my reading of the transition kernel suggests an induction proof should go through. However, because the cited thesis is unavailable to referees, the conditional verdict is appropriate. The concrete enumeration test I propose would settle whether the property actually fails for small n; if it passes, the remaining issue is purely one of proof availability, and the verdict should remain CONDITIONAL pending a written proof of Proposition 1.3. I therefore leave the reader's verdict unchanged.","tokens_in":57879,"tokens_out":29519,"duration_ms":275763,"concrete_test":"Perform an exact symbolic/numerical enumeration of all histories of the p-frozen model for n=5 and n=6, for p in {0.25, 0.5, 0.75, 1.0}. At every time m, compute the conditional distribution of the forest given (G_{p,n}(m), D_{p,n}(m)) and compare it with the uniform distribution on W(V_{p,n}(m), E_{p,n}(m)). If the total variation distance exceeds a tiny tolerance (e.g. 10^{-10}) for any reachable state, Proposition 1.3 is false for that p, and the jump estimates in Corollary 2.5, hence Theorem 1.2, are unsupported. If the check passes, it does not substitute for a proof, but it would shift the burden to a missing proof rather than a suspected counterexample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 1.2, is proved via Wormald's differential equation method. The trend estimates in Lemma 4.9 and the drift lower bounds in Lemma 4.4 use the jump asymptotics of Corollary 2.5, which in turn rely on Proposition 1.3: conditionally on (G_{p,n}(m), D_{p,n}(m)), the forest part is uniform over all forests with V_{p,n}(m) vertices and E_{p,n}(m) edges. For p=1/2 this is proved in [12], but for general p the paper cites Viau's unpublished thesis [34]. If Proposition 1.3 failed for some p in (0,1), then the expression in Proposition 2.1 for the probability that a given tree of size k freezes would not follow, Corollary 2.5 would be invalid, and the Wormald approximations underlying Theorem 1.2 would collapse. This is a genuine verification gap rather than a demonstrated contradiction: the transition kernel is shape-blind (merge, freeze, and tree-to-gel attachment rates depend only on component sizes), so an induction proof of Proposition 1.3 should be possible. The paper would be fully verifiable if that proof were included or if the thesis proof were made available.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the frozen Erdős-Rényi random graph F_{p,n}(m) for p in (0,1], establishing fluid limits for the gel size G_{p,n}, the number of discarded edges D_{p,n}, and forest statistics, as well as the asymptotic distribution of the total gelation time. The main result, Theorem 1.2, states that (G_{p,n}(⌊nt⌋)/n, D_{p,n}(⌊nt⌋)/n) converges in probability, uniformly on compacts, to an explicit deterministic pair (g_p(t), d_p(t)). The proof uses Wormald's differential equation method, with an epsilon-approximation to handle the non-Lipschitz behavior at t=1/2. The paper also proves fluid limits for the number of trees of each size, identifies the largest-tree asymptotics, and gives the limiting distribution of the absorption time in terms of a Gumbel variable and the digamma function. The proofs are detailed and the limit functions are defined explicitly, with no fitted parameters.","tokens_in":58138,"tokens_out":4722,"duration_ms":41789,"significance":"If the results are correct, the paper gives a complete law-of-large-numbers picture for a natural random graph model that has attracted recent attention, extending the p=1/2 results of Contat and Curien. The main strengths are the explicit, parameter-free limit functions, the careful handling of the non-Lipschitz critical point, and the concrete falsifiable predictions for the gelation time. The paper is also honest about its dependence on an unpublished thesis for a key structural input, which is the main verification concern. The overall proof strategy is credible and the technical work around the critical window appears sound, provided the cited free forest property holds for all p.","major_comments":[{"comment":"The free forest property is stated for all p in [0,1] and is cited to [12] for p=1/2 and to [34] (Viau's thesis, in preparation) for the general case. The paper describes the all-p generalization as 'without difficulty' but provides no proof, and the thesis is not yet available. This property is load-bearing: Proposition 2.1, Corollary 2.5, Lemma 2.6, Lemma 2.8, Lemma 4.4, and Lemma 4.9 all rely on it, and the Wormald-method proof of Theorem 1.2 collapses if the property fails. The manuscript should include a self-contained proof of Proposition 1.3 for all p in (0,1), or at least make the thesis proof publicly accessible, so that the central claim is verifiable.","section":"Proposition 1.3"}],"minor_comments":[{"comment":"In the proof, the notation '∫_0^1 (1−g_p(t)) dt' appears in the final display; the integral is over the variable s = g_p(t) and should be written as ∫_0^1 (1−s) … ds, or the left-hand side should read ∫_0^∞ (1−g_p(t)) dt. This is a typographical error but should be corrected.","section":"Lemma 3.2 proof"},{"comment":"The induction step in the proof of Theorem 1.6 is summarized as 'Details are left to the reader.' While Theorem 1.6 is a consequence of Theorem 1.2, the convergence of the ℓ1-valued sequence requires the induction to be filled in carefully, because the drift estimate (4.19) depends on the convergence of G_{p,n} and N^{(i)}_{p,n} for i≤k. Please expand this step.","section":"Section 4.5"},{"comment":"The notation F_{p,n} is used both for the graph process and, in the sentence 'let (F_{p,n}(m), m≥1) design the filtration generated by (G_{p,n},D_{p,n})', for the filtration. This is potentially confusing; a different symbol for the filtration would improve readability.","section":"Section 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically strong and the central result is plausible, but the proof of Theorem 1.2 depends critically on Proposition 1.3, whose all-p version is cited to an unpublished thesis. This is a genuine verification gap. I recommend asking the authors to include a proof of Proposition 1.3 in the paper or an appendix; the structure of the transition kernel suggests a direct induction should be possible. The induction in Theorem 1.6 should also be spelled out. No concerns about novelty or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does what the physics literature only sketched: it proves the fluid limit for the gel and discarded-edge processes in the frozen Erdős-Rényi model, and then derives precise asymptotics for the total gelation time and the extinction of trees of size k. The main theorems are genuinely new, and the proofs are detailed and serious. The handling of the non-Lipschitz problem at t = 1/2, by approximating the differential equations and then letting the approximation parameter tend to zero, is well executed. The results for the total gelation time, with the Gumbel fluctuations and the explicit digamma constants, are a real step beyond Krapivsky's heuristics.\n\nThe soft spot, as the stress-test note says, is Proposition 1.3: the free forest property for all p is cited to Viau's unpublished thesis. This is not a peripheral lemma. Lemma 4.9 and Lemma 4.4, and hence the whole Wormald argument for Theorem 1.2, depend on the jump asymptotics that come from it. For p = 1/2 the property is in the published Contat–Curien paper, but the all-p version is not. That is a genuine verification gap. I think it is probably fixable, because the transition kernel is shape-blind—merge and freeze rates depend only on component sizes—so an induction proof should be possible. But as it stands, a referee cannot fully verify the central theorem without access to the thesis. The induction step left to the reader in the proof of Theorem 1.6 is a much smaller issue; it is standard Grönwall plus the drift equation, and leaving it out is acceptable in a paper of this length.\n\nI do not see a circularity problem. The limit functions are defined by explicit ODEs, not fitted to the data. The self-citation to the thesis is structural input, not the target result. The paper is honest about the p = 0 case being open and about the unicycle component asymptotics being conjectural.\n\nWho should read this: anyone working on random graph processes, multiplicative coalescents, or the interface between probability and combinatorics. It deserves a serious referee. I would recommend sending it to peer review, with the instruction that the proof of Proposition 1.3—or at least a precise statement of where it appears in the thesis—be made available before final acceptance.","headline":"Rigorous fluid limits and gelation asymptotics for the frozen Erdős-Rényi model, resting on one load-bearing citation to an unpublished thesis.","tokens_in":58665,"tokens_out":1406,"would_cite":true,"duration_ms":17037,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C80","60C05","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every slowdown parameter $p\\in(0,1]$, the frozen random graph's gel and discarded-edge counts, rescaled by $n$, converge uniformly to explicit deterministic functions, and the total gelation time follows an extreme-value law with an…","keywords":["frozen Erdős-Rényi random graph","fluid limit","gelation","uniform random forests","differential equation method","total gelation time","extreme-value fluctuations","random graph phase transition"],"falsifier":"For a concrete test, simulate the frozen process for a fixed $p\\in(0,1)$ and moderately large $n$, and at a few times $t>1/2$ compare $G_{p,n}(\\lfloor nt\\rfloor)/n$ with $g_p(t)$ obtained by inverting $f_p$; the theorem asserts the difference goes to zero in probability, so a systematic nonzero discrepancy would falsify it. A more direct check of the premise is to test the free forest property itself at small $n$: condition on $G_{p,n}(m)$ and $D_{p,n}(m)$, sample the forest part, and compare its law with the uniform forest distribution on the same parameters; a reproducible deviation for any $p\\in(0,1)$ would invalidate the argument.","tokens_in":57668,"feed_emoji":"🧊","tokens_out":14704,"duration_ms":125521,"temperature":0.7,"pith_summary":"The paper proves a law of large numbers for a frozen random graph process: unicyclic components freeze as soon as they form, and slow down the absorption of further trees by a parameter $p$. For every $p\\in(0,1]$, the number of frozen vertices $G_{p,n}$ (the gel) and the number of discarded edges $D_{p,n}$, normalized by $n$ and read at time $nt$, converge in probability, locally uniformly in $t$, to deterministic functions $(g_p,d_p)$. The function $g_p$ is explicit: it is the inverse of $f_p(t)=\\frac{1}{2}+\\frac{t}{2p}\\int_0^1 \\frac{u^{1/p}}{1-tu}\\,du$, and $d_p(t)=t-g_p(t)-t(1-g_p(t))^2$. From this single fluid limit the paper derives the fluid limits of the forest part, the sizes of the largest trees, and the extreme-value asymptotics of the total gelation time---the first time all vertices are frozen. The upshot is that a small set of explicit formulas now describes the macroscopic evolution of a graph process that avoids the giant component, mirroring the classical description of the giant component.","feed_headline":"Frozen graph gel follows an explicit curve","feed_subtitle":"One integral's inverse determines the gel's growth, the forest's fate, and the gelation time.","key_machinery":"The central machinery is the free forest property plus the differential-equation method. Proposition 1.3 states that, conditionally on the forest part's vertex count and edge count, the forest is uniformly distributed over all labelled forests with those parameters. This turns every jump of the gel into a question about the combinatorics of uniform random forests: the conditional probability that a tree of size $k$ freezes is a ratio of forest counts, and those ratios are evaluated in the three regimes (subcritical, near-critical, supercritical) using the enumeration of random forests. The paper then applies the standard differential-equation method for discrete-time processes, but only on shifted time intervals $[1/2+\\varepsilon,\\infty)$ where the limiting vector field is smooth; the associated family of perturbed equations $(E(\\varepsilon))$ is controlled as $\\varepsilon\\downarrow 0$ and spliced together with bounds on the process up to time $n/2$, yielding Theorem 1.2.","core_discovery":"The central claim is Theorem 1.2: for fixed $p\\in(0,1]$, as $n\\to\\infty$ the pair $(G_{p,n}(\\lfloor nt\\rfloor)/n,\\ D_{p,n}(\\lfloor nt\\rfloor)/n)$ converges in probability, uniformly on compact time intervals, to $(g_p(t),d_p(t))$, where $g_p$ is the inverse of $f_p(t)=\\frac{1}{2}+\\frac{t}{2p}\\int_0^1 \\frac{u^{1/p}}{1-tu}\\,du$ and $d_p(t)=t-g_p(t)-t(1-g_p(t))^2$. Equivalently, on $(1/2,\\infty)$ the pair solves the explicit system of differential equations (E(0)) and its companion (\\widetilde{E}(0)) with zero initial condition at $t=1/2$. The paper calls this result the one on which all others rely; it completes earlier critical-window results by giving the supercritical regime, and for $p=1$ it recovers the classical fluid limit of the giant component. The rest of the paper develops the consequences: forest statistics, tree counts, largest trees, typical-tree geometry, and the gelation-time theorems.","pith_inferences":["The paper leaves implicit that the same explicit formulas, read as definitions, predict the $p=0$ fluid limit $g_0(t)=1-(2t)^{-1}$ and $d_0(t)=t-1+(4t)^{-1}$, with gelation time of order $n^2$ rather than $n\\ln n$; the paper only discusses this case as an open direction.","The uniform-forest route suggests that other constrained graph processes whose tree part is conditionally uniform could inherit the same differential-equation block, provided their jump kernels reduce to ratios of forest counts; this transfer is not made in the paper.","The Poisson limit for size-$k$ trees at the gelation threshold hints at a full process-level Poisson description of tree extinctions in the final phase, which the paper does not state.","The paper proves monotonicity of the fluid limit $g_p$ in $p$ even though the process itself has no stochastic monotonicity in $p$; a natural next question, not treated, is whether some partial stochastic order can be proven from the explicit formulas."],"forward_implications":["The gel mass grows from zero at $t=1/2$ with slope $2(1+p)$ and, at large times, $1-g_p(t)\\sim e^{-2pt}$, so the rescaled freezing process is initially fast and then exponentially slow.","For every $t\\neq 1/2$ the forest ratio $r_p(t)=t(1-g_p(t))$ is strictly below $1/2$, so the forest is subcritical: the $i$-th largest tree at time $nt$ is asymptotically $\\ln n/(2r_p(t)-1-\\ln(2r_p(t)))$.","The last tree of size $k$ disappears at time $\\frac{n}{2}(\\frac{\\ln n}{kp}+\\frac{k-1}{kp}\\ln\\frac{\\ln n}{kp})$ with extreme-value fluctuations, and the number of size-$k$ trees at that threshold converges to a Poisson law with explicit mean.","At $p=1$, the forest results re-derive classical facts for the standard random graph process, including the known connectedness-time asymptotics."],"supporting_citations":[{"why":"introduced the frozen random graph model and established the critical-window behavior and the free forest property for $p=1/2$, which the present paper generalizes and uses.","marker":"[12]"},{"why":"extends the free forest property and critical-window results to all $p\\in[0,1]$; the uniformity of the forest is the premise for the jump estimates.","marker":"[34]"},{"why":"provides asymptotic enumeration formulas for uniform random forests in the three regimes, used to evaluate the conditional jumps of the gel process.","marker":"[11]"},{"why":"gives the asymptotics of the largest components of uniform random forests, used for the largest-tree and typical-tree corollaries.","marker":"[22]"},{"why":"supplies the differential-equation method for discrete-time processes whose jumps are small and well approximated, adapted in the paper's proof of the fluid limit.","marker":"[36, 37]"},{"why":"presents the heuristic fluid-limit and gelation-time predictions in the physics literature that the paper states it confirms and completes.","marker":"[19]"}],"fun_headline_variants":["Frozen graph's gelation time now explicit","Explicit differential system drives frozen graph gel","Fluid limit of frozen Erdős-Rényi graph derived","Gelation curve emerges from frozen random graph model","Frozen graph gel follows a single inverse integral"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the free forest property---that, conditionally on the forest's vertex and edge counts, the forest part is uniformly distributed over labelled forests with those counts---which is cited to [12] for $p=1/2$ and to an in-preparation thesis [34] for general $p$, and which supplies the jump and drift estimates on which the differential-equation argument depends.","fun_headline_variants_meta":{"raw":{"variants":["Frozen graph's gelation time now explicit","Explicit differential system drives frozen graph gel","Fluid limit of frozen Erdős-Rényi graph derived","Gelation curve emerges from frozen random graph model","Frozen graph gel follows a single inverse integral"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000175,"raw_usage":{"total_tokens":1309,"prompt_tokens":989,"completion_tokens":320,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":246}},"tokens_in":605,"tokens_out":320,"duration_ms":3462,"temperature":1.0,"reasoning_tokens":246,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T15:21:37.719584+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete test, simulate the frozen process for a fixed $p\\in(0,1)$ and moderately large $n$, and at a few times $t>1/2$ compare $G_{p,n}(\\lfloor nt\\rfloor)/n$ with $g_p(t)$ obtained by inverting $f_p$; the theorem asserts the difference goes to zero in probability, so a systematic nonzero discrepancy would falsify it. A more direct check of the premise is to test the free forest property itself at small $n$: condition on $G_{p,n}(m)$ and $D_{p,n}(m)$, sample the forest part, and compare its law with the uniform forest distribution on the same parameters; a reproducible deviation for any $p\\in(0,1)$ would invalidate the argument.","supporting_citations":[{"cited_title":"Contat and N","cited_arxiv_id":null,"evidence_quote":"introduced the frozen random graph model and established the critical-window behavior and the free forest property for $p=1/2$, which the present paper generalizes and uses."},{"cited_title":"Viau, Graphes d’Erdős-Rényi gelés, PhD thesis, Université Sorbonne Paris-Nord, (in prepa- ration)","cited_arxiv_id":null,"evidence_quote":"extends the free forest property and critical-window results to all $p\\in[0,1]$; the uniformity of the forest is the premise for the jump estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides asymptotic enumeration formulas for uniform random forests in the three regimes, used to evaluate the conditional jumps of the gel process."},{"cited_title":"Łuczak and B","cited_arxiv_id":null,"evidence_quote":"gives the asymptotics of the largest components of uniform random forests, used for the largest-tree and typical-tree corollaries."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"presents the heuristic fluid-limit and gelation-time predictions in the physics literature that the paper states it confirms and completes."}],"review_version":1}