REVIEW 4 major objections 4 minor 42 references
The contact process on Scale-Free Percolation
T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read Even sparse power-law networks with finite-variance degrees keep the contact process alive for exponentially long times.
desk verdict First exponential-extinction result for contact process on scale-free percolation in the finite-variance small-world regime; the main theorem is plausible but rests on an unproved growing-parameter generalization of the constellation lemma. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the $(S,D,\Delta)$-constellation: a tree subgraph with distinguished vertices (stars) of degree at least $S/2$, consecutive stars at graph distance at most $D$, and the star-to-star skeleton itself a tree of degree at most $\Delta$. Proposition 2.1 extends a theorem of Mountford, Mourrat, Valesin and Yao by allowing $S$ and $D$ to grow with $n$; it asserts that if $S_n\ge C\lambda^{-2}\log(1/\lambda)D_n$, then the contact process on the constellation survives for time at least $e^{c(\lambda^2 S_n+|J_n|)}$ with high probability. The proof of part (ii) uses a multiscale partition of the box into nested cubes of side $(\log n)^{\theta^k A}$, choosing the highest-weight vertex in each coarse box as a potential star, controlling the largest connected components of fine subcubes via supercritical percolation estimates, and connecting the stars by paths of length $O((\log n)^{\nu_p})$ using a red/blue chessboard colouring to keep the paths disjoint. The constellation has $|J_n|\ge c n (\log n)^{-A}$ stars, which yields the exponential-with-log-correction survival time; in the ultra-small-world regime, heavier tails allow a simpler construction with $\Theta(n)$ stars at distance $1$, giving survival time $e^{c n}$.
What would settle it
For fixed $d=1$, $\alpha=1.5$, $\tau=4$ (so $\gamma=4.5>2$, $\alpha\in(d,2d)$) and a small $\lambda$, estimate the median extinction time from full occupancy on scale-free percolation boxes for increasing $n$. The theorem predicts $\log\tau_{G_n}\ge c n(\log n)^{-A}$ for every $A>2\gamma/(2-\alpha/d)$, so if $\log\tau_{G_n}/(n(\log n)^{-A})$ is observed to go to $0$ for a range of $A$, the claimed rate is false.
Extended reading notes
Core claim
In the model, vertices come from a unit-intensity Poisson process on $[0,n^{1/d})^d$, each vertex gets a Pareto weight with tail index $\tau-1$, and two vertices $x,y$ are joined independently with probability $1-\exp(-\rho W_x W_y/\|x-y\|^\alpha)$. Writing $\gamma=\alpha(\tau-1)/d$, the degree distribution has power-law exponent $\beta=\gamma+1$. The paper's main theorem states that for $\alpha>d$ and $\rho$ above the percolation threshold: (i) if $\gamma\in(1,2)$, then for every $\lambda>0$ there is $c>0$ with $\mathbb{P}(\tau_{G_n}\ge e^{c n})\to 1$; (ii) if $\gamma>2$ and $\alpha\in(d,2d)$, then for every $\lambda>0$ and every $A>2\gamma/(2-\alpha/d)$ there is $c>0$ with $\mathbb{P}(\tau_{G_n}\ge \exp(c n (\log n)^{-A}))\to 1$. Part (ii) is the genuinely new result: the graph is supercritical but sparse, with finite-variance degrees and only logarithmic graph distances, and the proof finds a constellation subgraph containing polylogarithmically many stars that lets the infection persist. The paper also states a matching-order result for the non-extinction probability $\Gamma(\lambda)$ as $\lambda\to 0$ in the regime $\gamma\in(1,2)$, recorded as Theorem 1.2.
Load-bearing premise
The generalized star-to-star infection estimate of Proposition 2.1 is the load-bearing step: it assumes that a star of size $S_n$ can keep its neighbourhood sufficiently occupied and pass the infection to stars at distance $D_n$, with $S_n$ and $D_n$ growing polylogarithmically, even though the lemmas it is built on were originally proved for bounded sizes and distances; the paper says only that the generalization requires minor changes and does not spell them out.
Editorial extensions
If this is right
- For every $\lambda>0$, the contact process on finite scale-free percolation boxes is metastable: starting from full occupancy, extinction is exponentially unlikely in both the infinite-variance and finite-variance degree regimes.
- In the finite-variance small-world regime, the lower bound is $\exp(c n (\log n)^{-A})$ with $A>2\gamma/(2-\alpha/d)$; the logarithmic factor is a residue of the star-counting construction, and the authors leave open whether the true rate is fully exponential.
- In the ultra-small-world regime ($\gamma\in(1,2)$), the theorem holds for all $\rho>0$, since $\rho_c=0$, so no supercritical percolation assumption is needed there.
- The proof identifies an explicit random substructure — many high-degree stars joined by disjoint short paths — that supports the infection; any graph containing such a constellation inherits the exponential survival lower bound.
- The non-extinction probability $\Gamma(\lambda)$ from a single infected vertex obeys the same power-law order as configuration models and hyperbolic graphs in the $\gamma\in(1,2)$ range, with a logarithmic correction when $\gamma\in(3/2,2)$.
Reading between the lines
- If the logarithmic correction is genuinely necessary rather than a proof artefact, the phase transition at $\beta=3$ (the finite-variance threshold) would mark a qualitative change in metastability rates: survival time would drop from $e^{c n}$ to $e^{c n/(\log n)^A}$, a slower but still exponential scale. This prediction is not tested in the paper.
- A natural testable extension is to simulate the contact process on scale-free percolation for parameters $d=1$, $\alpha=1.5$, $\tau=4$ (so $\gamma=4.5>2$) and check whether $\log\tau_{G_n}\cdot(\log n)^A/n$ settles at a positive constant; if it tends to zero, the lower bound is not tight.
- The same constellation construction might carry over to geometric inhomogeneous random graphs and other kernel-based spatial graphs in their finite-variance regime, because the proof uses only the Pareto weight tail and the polynomial connection kernel, though the paper does not state this.
- The Theorem 1.2 asymptotics imply that in the ultra-small-world regime, the survival probability from a single vertex vanishes polynomially in $\lambda$, with the same exponents as non-spatial power-law configuration models; this suggests the spatial embedding does not change the critical exponent there, a comparison the paper sketches but does not develop.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the extinction time of the contact process on scale-free percolation (SFP) restricted to a box of volume n in R^d. In the ultra-small-world regime γ=α(τ−1)/d∈(1,2), it claims an exponential lower bound exp(cn) on the extinction time, adapting the hyperbolic-random-graph method of Linker et al. In the small-world finite-variance regime γ>2 with α∈(d,2d), the main contribution, it claims a lower bound of order exp(c n (log n)^{−A}) for every A>2γ/(2−α/d), valid for every λ>0. The proof combines a generalized constellation theorem (Proposition 2.1), a multi-scale construction of a polylogarithmic constellation in SFP (Proposition 3.1), and percolation-type estimates for component sizes and connecting paths. The paper also states, as Theorem 1.2, asymptotic results for the non-extinction probability Γ(λ) in the regime γ∈(1,2), with only a proof sketch in Section 5.
Significance. If the main theorem is correct, it is a substantial contribution to the metastability theory of the contact process on spatial random graphs. The finite-variance, small-world regime has not been treated before for SFP, and the polylogarithmic correction to the exponential rate is a genuinely new feature relative to the ultra-small-world case. The multi-scale construction in Section 3 is technically ambitious and goes well beyond a direct extension of earlier work. The paper is also transparent about the limits of its method, notably in Remark 1.3. However, the central mechanism Proposition 2.1 is not proved in full; it is delegated to a 'minor changes' generalization of cited results, and this is the main bottleneck for verifying the paper's headline claim.
major comments (4)
- [Section 2.2, Lemma 2.2] Proposition 2.1 is the mechanism that converts the polylogarithmic constellation of Proposition 3.1 into the survival lower bound of Theorem 1.1(ii), but its proof is not self-contained at the critical point. Lemma 2.2 asserts (17) and (18) as 'a direct application' of [Mountford et al., 2013, Lemmas 3.1(ii) and 3.2] and [Mountford et al., 2016, Lemma 6.2], adding only that 'careful analysis of the proof shows' the infested-star estimate and that 'the proof of this generalisation only requires minor changes'. The cited statements are for fixed star size and fixed inter-star distance, whereas Theorem 1.1(ii) requires S_n and D_n to grow polylogarithmically with n. In particular, the text does not show how D_n enters the failure exponent in (18) or why the union bound in Lemma 2.3 continues to hold with a constant independent of n once D_n grows. Since condition (10) is only S_n ≥ C λ^{-2} log(1/λ) D_n, a hidden dependence of the failure probability on D_n would destroy the argument. This missing generalization is load-bearing and must be supplied in full.
- [Section 3.6, Eq. (44)] The inequality P(R_i and R_{i+1} are not connected | E1∩E2) ≤ P(R_i and R_{i+1} are not connected | E1) is justified by FKG, with the argument that the first event is decreasing and E2 is increasing. However, the conditioning is on E1∩E2, not on E2, and E1 is the intersection of a lower-tail event (increasing) and an upper-tail event (decreasing) for component sizes, so E1 is not monotone. The FKG inequality for increasing events therefore does not directly give the displayed bound. The same issue appears in the derivation of (43). Since (45) and the subsequent union bound control E0, this is a substantive gap in Lemma 3.4 and needs an additional argument, for example a conditional FKG statement or replacement of E1 by a monotone event.
- [Section 3.5, Eqs. (36)-(38)] The proof of Lemma 3.3 bounds the probability of the success event E_2^2(i) = {bx_i ∈ C^f_{f(i)}} rather than its complement. Equation (36) gives an upper bound for P(E_2^2(i) | E_1^2∩E_1), and the estimate (38) makes this upper bound small; this would show that bx_i belongs to the largest component with small probability, which is the opposite of what Lemma 3.3 needs. The final union-bound display in the lemma then uses (36) as if it controlled P(E_2^2(i)^c | ...). One can repair the argument by applying the same union-bound reasoning to the complement event, but as written the lemma does not establish P(E_2^2|E1∩E1_2)→1.
- [Section 5] Theorem 1.2 is stated as a theorem, but the section is explicitly a sketch. The adaptation of [Linker et al., 2021, Theorem 1.1] to SFP is described qualitatively, and the four lemmas behind the upper and lower bounds in the two regimes are not proved; in particular, the treatment of soft edges via (54) and (56) is not accompanied by the path-counting estimates or the star-construction lemma needed for the upper bounds. If Theorem 1.2 is to remain part of the paper, its proof must be completed; alternatively, the statement should be reclassified as a conjecture or clearly separated from the main proved theorem.
minor comments (4)
- [Section 3.6, before Eq. (44)] The display 'P(E0|E1∩E2) = 1 − P(E0|E1∩E2)' should read 'P(E0^c|E1∩E2)', since the printed formula is a tautology and does not express the intended complement probability.
- [Section 5] The phrase 'For,r >1' is a typo and should read 'For r>1'.
- [Abstract and Theorem 1.1] The abstract says β≥3, while Theorem 1.1(ii) is stated for γ>2, i.e. β>3; the boundary case β=3 is not covered and the wording should be aligned.
- [Remark 1.3] The Schapira–Valesin bound is missing a closing parenthesis in the displayed inequality; as printed it is not a well-formed formula.
Circularity Check
No significant circularity: the main extinction-time bound is derived from an explicit structural construction in SFP plus external contact-process estimates, not from a fitted or self-referential input.
full rationale
The derivation chain is self-contained in the sense required here. Theorem 1.1(ii) is proved by (a) proving Proposition 3.1, which constructs, with probability tending to 1, a subgraph of the SFP restriction containing |J_n| ≥ c1 n (log n)^{-A} stars of degree at least c2(log n)^{\nu_s} connected by disjoint paths of length at most c3(log n)^{\nu_p}; and (b) applying Proposition 2.1, a constellation-level lower bound on the contact-process extinction time whose proof is adapted from the external results of Mountford, Valesin and Yao (2013, 2016). The constants are chosen to satisfy explicit inequalities (\nu_s > \nu_p and A > 2\gamma/(2-\alpha/d)), and the final bound P(\tau_{G_n} \ge \exp(c n (\log n)^{-A})) \to 1 follows by substitution, not by construction from the conclusion. No parameter is fitted to the claimed extinction time, and no prediction is a rewritten input. The self-citations (Dalmau and Salvi 2021; Cipriani and Salvi 2024) are used only as background on SFP degrees and mixing times and are not load-bearing for the lower bound. The paper explicitly flags one unproved generalization inside the proof of Lemma 2.2: it says the adaptation of Mountford et al.'s lemmas 'only requires minor changes' and, in one place, 'We skip the details'. That is a rigor gap in the proof of Proposition 2.1, not a circular reduction: the cited star and path estimates are external facts that either hold or fail independently of the paper's target theorem. The same applies to the noted FKG conditioning step in Lemma 3.4 and the inequality in Lemma 3.3; these are internal technical concerns, not self-referential definitions. Accordingly, no circular step satisfying the quoted-reduction test is present.
Assumptions & free parameters
assumptions (6)
- standard math FKG inequality holds for increasing events in the Poissonian construction of SFP (Gracar et al. 2022).
- domain assumption Deprez and Wüthrich (2019), Theorem 3.4: in a box of side L, the largest connected component of supercritical SFP is of order L^d with probability at least 1-exp(-c L^{d(1-α'/d)}) for α'∈(α,2d).
- domain assumption Deprez and Wüthrich (2019), Theorem 3.2: ρ_c=0 for γ∈(1,2), so any ρ>0 is supercritical.
- domain assumption Mountford, Valesin and Yao (2013), Lemmas 3.1 and 3.2, and Mountford et al. (2016), Lemma 6.2: a star of size S retains infection for time exp(c λ^2 S) and can infect a neighboring star at distance D if S ≥ C λ^{-2} log(1/λ) D.
- domain assumption Mountford et al. (2016), Proposition 5.2: the contact process on a tree with bounded degree Δ survives for time exp(c |J|) starting from full occupancy.
- domain assumption Dalmau and Salvi (2021), Theorem 2.2 and Proposition 3.3: the degree tail of SFP is a power law with exponent γ and the expected degree of a vertex of weight w is Θ(w^{d/α}).
Cite this review
Pith. "Pith review of The contact process on Scale-Free Percolation." pith.science (2026). https://pith.science/paper/N4DAIUKF
@misc{pith2026250510582,
author = {Pith},
title = {Pith review of: The contact process on Scale-Free Percolation},
year = {2026},
howpublished = {\url{https://pith.science/paper/N4DAIUKF}},
note = {Machine review of arXiv:2505.10582}
}
abstract
We consider the contact process on scale-free percolation, a spatial random graph model where the degree distribution of the vertices follows a power law with exponent $\beta$. We study the extinction time $\tau_{G_n}$ of the contact process on the graph restricted to a d-dimensional box of volume n, starting from full occupancy. In the regime $\beta \in (2, 3)$, where the degrees have finite mean but infinite variance and the graph exhibits the ultra-small world behaviour, we adapt the techniques of [Linker et al., 2021] to show that $\tau_{G_n}$ is exponential in n. Our main contribution, though, deals with the case $\beta \geq 3$, where the degrees have finite variance and the graph is small-world. We prove that also in this case $\tau_{G_n}$ grows exponentially, at least up to a logarithmic correction reflecting the sparser graph structure. The proof requires the generalization of a result from [Mountford et al., 2016] and combines a multi-scale analysis of the graph, the study of the chemical distance between vertices and percolation arguments.
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