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Fluid limit and gelation in the frozen Erd\H{o}s-R\'enyi random graph

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Pith's one-line read 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…

desk verdict Rigorous fluid limits and gelation asymptotics for the frozen Erdős-Rényi model, resting on one load-bearing citation to an unpublished thesis. read the letter →

arxiv 2502.01424 v1 pith:SIRLXTYJ submitted 2025-02-03 math.PR

Bénédicte Haas , Vincent Viau This is my paper · ORCID
classification math.PR MSC 05C8060C0560F05
keywords frozenErdős-Rényirandomgraphfluidlimitgelationuniformforestsdifferentialequationmethodtotaltimeextreme-valuefluctuationsphasetransition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 3 minor

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.

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 (1)
  1. [Proposition 1.3] 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.
minor comments (3)
  1. [Lemma 3.2 proof] 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.
  2. [Section 4.5] 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.
  3. [Section 2] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the fluid limit is verified against explicit ODEs, and the only self-cited input (free forest property for all p, cited to a thesis in preparation) is a verification gap rather than a circular reduction.

full rationale

The derivation of Theorem 1.2 is self-contained in the sense that matters for circularity: the claimed limit functions g_p and d_p are defined before any asymptotics (Definition 1.1 and Eq. 1.3) by an explicit inverse function and an algebraic relation; the proof via Wormald's differential equation method computes conditional jump expectations from the model's transition rules and Britikov's forest enumeration, and then verifies that the approximating ODEs converge to the defining equations (E(0)) and (˜E(0)). There are no fitted parameters and no step in which the target convergence is assumed to define the limit. The subsequent results (Theorems 1.6, 1.9, and the corollaries) are derived from Theorem 1.2, which is the announced logical dependence, not a circular one. The only flagged item is Proposition 1.3: the free forest property for general p is stated with citation to [12] for p=1/2 and to the co-author's thesis [34] for all p, and it is then used through Proposition 2.1 and Corollary 2.5 in Lemmas 4.4 and 4.9. This is a genuine verification gap because [34] is in preparation, but it is not a circular reduction: the property is a structural input about the model's conditional law, not the fluid-limit conclusion, and the published p=1/2 case provides independent support for the mechanism. The paper would be fully checkable if the proof of Proposition 1.3 for all p were included, but absent that, the central fluid-limit claim still has independent content and does not reduce to its own assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the model dynamics plus several classical results (Wormald's theorem, Britikov's forest asymptotics, Luczak-Pittel component sizes) and on two structural inputs from prior work: the free forest property and the critical-window estimates of [12,34]. No parameters are fitted to data; p is an exogenous model parameter.

assumptions (5)
  • standard math Wormald's differential equation method (Theorem A.4)
    The main technical engine for the fluid-limit proofs; the paper adapts it to handle the non-Lipschitz critical point.
  • standard math Britikov/Kolchin asymptotic enumeration of uniform random forests (Propositions 2.2, 2.3)
    Used to estimate conditional jump probabilities via the free forest property in Section 2.
  • standard math Luczak-Pittel largest-tree estimates for uniform forests (Proposition 2.4)
    Used to prove Corollary 1.7 on the largest trees and to control jump sizes.
  • domain assumption Free forest property for all p in [0,1] (Proposition 1.3)
    Structural fact about the model: conditionally on G and D, the forest is uniform. The p=1/2 version is published in [12]; the all-p generalization is cited to Viau's thesis [34].
  • domain assumption Critical-window scaling results from [12,34] (gel and discarded edges are O_P(n^{2/3}) and o_P(n^{2/3}) at time n/2)
    Used in (4.1) to start the fluid-limit proof beyond the critical window.

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Cite this review

Pith. "Pith review of Fluid limit and gelation in the frozen Erd\H{o}s-R\'enyi random graph." pith.science (2026). https://pith.science/paper/SIRLXTYJ

@misc{pith2026250201424,
  author       = {Pith},
  title        = {Pith review of: Fluid limit and gelation in the frozen Erd\Hos-R\'enyi random graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SIRLXTYJ}},
  note         = {Machine review of arXiv:2502.01424}
}
abstract

The frozen Erd\H{o}s-R\'enyi random graph is a variant of the standard dynamical Erd\H{o}s-R\'enyi random graph that prevents the creation of the giant component by freezing the evolution of connected components with a unique cycle. The formation of multicyclic components is forbidden, and the growth of components with a unique cycle is slowed down, depending on a parameter $p\in [0,1]$ that quantifies the slowdown. At the time when all connected components of the graph have a (necessary unique) cycle, the graph is entirely frozen and the process stops. In this paper we study the fluid limit of the main statistics of this process, that is their functional convergence as the number of vertices of the graph becomes large and after a proper rescaling, to the solution of a system of differential equations. Our proofs are based on an adaption of Wormald's differential equation method. We also obtain, as a main application, a precise description of the asymptotic behavior of the first time when the graph is entirely frozen.

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