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Oriented bond-site percolation in random environment and contact processes with periodic recovery

T0 review · 1 major / 8 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A $(1+\varepsilon)$-moment condition on column stretches is enough to force percolation at large $\lambda$, and the result implies survival of contact processes with periodic recovery.

desk verdict Genuine extension of the stretched-lattice phase transition to oriented bond-site percolation, with new contact-process applications; the theorem's condition (ii) is misstated for λ=0, but the fix is local and the paper deserves a serious referee. read the letter →

arxiv 2507.00329 v1 pith:ATBVLGRC submitted 2025-06-30 math.PR

Benedikt Jahnel , Lukas Lüchtrath , Anh Duc Vu This is my paper
classification math.PR MSC 60K0560K3582B43
keywords orientedpercolationrandomenvironmentcolumnardisorderrandomlystretchedlatticecontactprocessperiodicrecoverygeneralisedmultiscalerenormalisation
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 establishes a percolation phase transition for oriented bond-site percolation on the planar lattice when entire spatial columns are randomly stretched: as long as the stretches have a finite $(1+\varepsilon)$-th moment and the connection functions satisfy a mild exponential lower bound, there is a finite critical value $\lambda_c$ such that percolation occurs for every $\lambda>\lambda_c$, almost surely in the environment. This extends earlier stretched-lattice results to a mixed bond-site setting with columnar disorder, where dependencies along columns do not decorrelate. The authors then couple this percolation model to generalised contact processes and prove survival for sufficiently large infection rates when recoveries are periodic with a random shift, and for small recovery rates in a random spatial environment with unbounded intensities. A phase transition follows because earlier extinction results cover small $\lambda$. The paper also shows that temporal stretches with heavier-than-exponential tails remove the phase transition entirely.

What carries the argument

The load-bearing object is the static multiscale renormalisation scheme: a stationarised renewal embedding of the environment with interarrival times $\xi_i+\nu_{i,i+1}$, a hierarchy of blocks whose sizes grow super-exponentially ($L_k=L_{k-1}\lfloor L_{k-1}^{\gamma-1}\rfloor$), and a definition of good blocks that permits at most one bad sub-block (or two consecutive bad sub-blocks). Lemma 2.4, taken from the stretched-lattice literature, uses the $(1+\varepsilon)$-moment condition through a decoupling inequality to bound the probability of a bad block by $L_k^{-\alpha}$. On good rectangles, Lemmas 2.7 and 2.8 show that left-right, right-left, and bottom-top crossings fail with probability at most $\exp(-L_k^\beta)$; planarity then patches these crossings into an infinite path. The connection-function condition $\exp(-\sigma s)\le \kappa_\lambda(s)$ is the one that lets even the longest stretches be crossed with probability bounded away from zero once $\lambda$ is large.

What would settle it

Compute the covariance of the consecutive stretches $\nu_{i,i+1}=-\log(2\,d_T(U_i,U_{i+1}))$ in the continuous periodic-recovery case: because $U_i$ and $U_{i+1}$ are shared between neighbouring stretches, the covariance is nonzero, which shows the environment is not i.i.d.; alternatively evaluate $\kappa_0(1)=0$ in that example, which violates the theorem's condition $\exp(-\sigma)\le \kappa_\lambda(1)$ for every $\lambda\ge0$.

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Extended reading notes

Core claim

The central claim, Theorem 1.2, is that OPRE percolates under the sole moment condition $\mathbb{E}[\xi^{1+\varepsilon}], \mathbb{E}[\nu^{1+\varepsilon}]<\infty$ and connection functions that are monotone, satisfy $\exp(-\sigma s)\le \kappa_\lambda(s)$ with $\kappa_\lambda(s)\to 1$ as $\lambda\to\infty$: there exists $\lambda_c<\infty$ such that for every $\lambda>\lambda_c$ the model contains an infinite open path for almost every environment. The proof embeds the stretched columns stationarily as a renewal process, partitions space into scales of super-exponentially growing blocks, declares blocks good unless they contain too many bad sub-blocks, and bounds horizontal and vertical crossing failures by $\exp(-L_k^\beta)$. The applications are the paper's payoff: for the contact process with periodic recoveries of the form $2(\mathbb{Z}+U)$ or $2\mathbb{Z}+B$ on the line, survival occurs for all sufficiently large $\lambda$; and for the contact process in a random environment with unbounded recovery intensities, survival is proved for small $\delta$ under only a $(1+\varepsilon)$-moment condition on the high intensity.

Load-bearing premise

The argument assumes that the random stretches in different columns are independent and have a finite $(1+\varepsilon)$-th moment, yet in the periodic-recovery coupling consecutive stretches share a random uniform and are therefore dependent, and the required exponential lower bound is verified only for large $\lambda$.

Editorial extensions

If this is right

  • The contact process with periodic recovery in both the continuous shift and Bernoulli shift cases has a non-trivial phase transition: extinction for $\lambda\le(4d)^{-1}$ and survival for all sufficiently large $\lambda$.
  • The contact process in a random spatial environment survives for every $\delta$ below a positive critical value, even when the high recovery intensity has only a finite $(1+\varepsilon)$-moment.
  • Any family of connection functions satisfying monotonicity and the exponential lower bound yields OPRE percolation for large $\lambda$, so the theorem applies uniformly across vertex and edge environments.
  • For oriented percolation with temporal stretches of heavier-than-exponential tails, no phase transition exists: percolation fails for every $p\in(0,1)$.
  • The $(1+\varepsilon)$-moment assumption enters only through the decoupling inequality, leaving open whether finite first moments would suffice.

Reading between the lines

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

  • If the decoupling inequality can be proved under weaker dependence, the same scheme should apply to environments with short-range correlations between columns; a concrete test is to run the bad-block estimate on moving-average stretches.
  • The CPPR coupling suggests that survival should persist when periodic recoveries are replaced by almost-deterministic interarrival times with bounded support, since only the moments of the stretch variables enter the OPRE conditions.
  • The paper's time-limitation mechanism suggests a quantitative prediction for the Bernoulli-shift case: crossing a block of length $k$ costs about $\lambda^k/k!$, so the critical infection rate should scale roughly linearly with $k$, which is testable by simulation.
  • The complementary periodic-infections setting, which the paper leaves open, is a natural candidate for the same coupling and may exhibit a non-trivial extinction phase in one dimension.
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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 / 8 minor

Summary. The paper studies oriented bond-site percolation on the planar lattice L2 with columnar random stretches (ξ_x) and (ν_{x,x+1}) and a family of connection functions κλ. Theorem 1.2 asserts that, under a (1+ε)-moment condition on the stretches and certain assumptions on κλ, the model percolates for all sufficiently large λ. The proof uses a static multiscale renormalisation scheme adapted from Hilário et al. [12]. The result is applied to contact processes with periodic recovery (continuous and discrete random shifts) and to contact processes in a random spatial environment, yielding survival for large infection rates; the paper also proves absence of percolation for temporal stretches with heavier-than-exponential tails (Proposition 1.3).

Significance. If Theorem 1.2 is taken with the corrected hypothesis on λ indicated below, the paper is a solid contribution: it generalises the renormalisation scheme of [12] from undirected percolation to oriented bond-site percolation, and the couplings to the contact-process applications are explicit and checkable. The moment condition E[ξ^{1+ε}], E[ν^{1+ε}]<∞ is natural, and the applications give non-trivial survival/extinction phase transitions when combined with the extinction result of [7]. The paper contains no fitted parameters or circular predictions; the main technical input from [12] is external and published. Proposition 1.3 is a clean, correct observation. The main defect is in the statement of condition (ii) of Theorem 1.2, which does not match the paper's own examples; this is local and fixable. The dependence concern raised in the stress-test note about Case (Uni) is, in my reading, not an actual problem.

major comments (1)
  1. [Theorem 1.2, Eq. (3)] Condition (ii) as stated requires exp(-σs) ≤ κλ(s) for every λ≥0, but this is false for the paper's own examples: in Section 2.1.1, κλ(s)=1-exp(-λe^{-s}) and in Section 2.1.2, κλ(s)=P(A_⌊√s⌋), and both give κ_0(s)=0 for every s, so no σ>0 can exist. The verification in Section 2.1.1 proves the lower bound only for λ≥2 (Eq. (6)) and handles s≤s0 only for sufficiently large λ; Section 2.1.2 explicitly assumes λ≥100. The renormalisation proof (Lemma 2.6 and Section 2.3) needs the lower bound only for the single large λ at which the scheme is run, not uniformly for all λ≥0. I recommend changing condition (ii) to require the exponential lower bound for all λ≥λ0 for some λ0≥0 (equivalently, for all sufficiently large λ), and adjusting the applications to verify this with their respective λ0. This is a local fix and the main proof is unaffected.
minor comments (8)
  1. [Section 2.1.1, Case (Uni)] The stress-test concern about dependence of the stretches in Case (Uni) does not land: conditionally on U_i, S_i=2d_T(U_i,U_{i+1}) is uniform on [0,1] and independent of (U_0,...,U_{i-1}), and the resulting sequence (S_i) is i.i.d. uniform; hence ν_i=-log S_i forms an i.i.d. exponential environment as required by Theorem 1.2.
  2. [Section 2.2.2, definition of q_k] The symbol DTC in the definition of q_k just before Lemma 2.6 should be BTC, the bottom-top crossing event defined in the same section; the right-left crossing RLC was defined, but there is no DTC event.
  3. [Section 2.1.2, Case (Ber)] In the text 'set ν := K 2', the notation should read ν := K^2; the identity P(A_K)=κλ(ν) is only correct with ν=K^2, since κλ(s)=P(A_⌊√s⌋).
  4. [Section 2.2.3, Eq. (18)] In the lower bound for traversing a bad area, the displayed factors should be e^{-ν_{x,x+1}} e^{-ξ_x}, not e^{ν_{x,x+1}} e^{ξ_x}; as written the inequality is false because κλ(s) is bounded above by 1, not by e^s.
  5. [Section 1.2, paragraph on periodic recovery] There is a duplicated phrase 'this argument this argument' in the discussion of almost deterministic interarrival times; please correct the typo.
  6. [Section 2.2.1, Lemma 2.4] Lemma 2.4, the decoupling estimate P(I_{k,i} is bad) ≤ L_k^{-α}, is stated without proof and attributed to [12, Lem. 3.1]; since Theorem 1.2's induction in Lemma 2.6 and the Borel–Cantelli argument at (19) rely on it, please state the decoupling inequality used and explain why it transfers verbatim to the present bond-site setting, or include the proof.
  7. [Section 2.1.3, Proposition 1.7] The sum in the definition of ν (N_0 log N_0 + ∑_{\ell=0}^{N_0} Δ_\ell) does not match the exponent in Eq. (10), which sums over \ell=X_b+1,\dots,X_{b+1}-1; please make the stochastic domination explicit (for instance by taking ν = N log N + ∑_{i=1}^{N} Δ'_i with independent copies) so that κ_L(ν) ≤ P(edge open) holds.
  8. [Proof of Theorem 1.2, definition of H_k] In the definition of H_k, the events LRC(R_ver_k(i,i)) and BTC(R_hor_k(i,i+1)) appear interchanged; based on the rectangle definitions (15)–(16), H_k should involve LRC(R_hor_k) and BTC(R_ver_k) for the patching argument in Figure 13 to be correct.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: Theorem 1.2 rests on external renormalisation from [12]; the CPPR applications are honest couplings. The condition-(ii) mismatch for all λ≥0 is a correctness gap, not a circular step.

full rationale

I walked the derivation chain of Theorem 1.2 and its applications. The main proof is explicitly based on the static multiscale renormalisation scheme of [12], an external paper with no author overlap with the present manuscript. The decoupling bound in Lemma 2.4 is cited to [12, Lem. 3.1] with the proof omitted because it is identical; this is independent support, not a self-citation. The applications in Section 2.1 are genuine couplings: they construct OPRE environments and connection functions from the contact-process data and translate percolation into survival. No parameter is fitted to a subset of data and then renamed a prediction; for example, in Case (Uni), κλ(s)=1-exp(-λe^{-s}) and ν=-log(S) are chosen so that κλ(ν) equals the edge-open probability 1-exp(-λS), and Theorem 1.2 then does the work. The reader's dependence concern about Case (Uni) does not establish circularity: for i.i.d. uniforms U_i, the variables S_i=2d_T(U_i,U_{i+1}) are i.i.d. uniform on [0,1], so the OPRE environment is i.i.d. as required. The genuine issue is a hypothesis mismatch: condition (ii) of Theorem 1.2 demands exp(-σs)≤κλ(s) for every λ≥0, but the paper's own examples have κ_0=0 and the proof verifies the lower bound only for large λ (λ≥2 in Case (Uni), λ≥100 in Case (Ber)). This affects correctness of the stated theorem or its application, but it is not circular: the applications do not assume the desired survival conclusion. The self-citations [9], [15], [16], and [17] are background or motivational; [15] is used only as informal intuition for exponential survival to motivate rectangle heights and is not load-bearing. Overall, the derivation is self-contained against an external benchmark and shows no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No parameters are fitted to data; the proof uses abstract scale parameters (ε, α, γ, L0, µ, β) chosen within inequalities. The main external input is the decoupling/bad-block lemma from [12]. No new physical entities are introduced.

assumptions (3)
  • domain assumption Decoupling inequality for i.i.d. environments with (1+ε) moments (Lemma 2.4, quoted from [12, Lem. 3.1]).
    Used to bound probabilities of bad blocks in the multiscale renormalization; the proof is not included in this paper.
  • standard math Ergodicity of the environment equates almost-sure percolation with positive probability of percolation from the origin.
    Invoked after Theorem 1.2 to justify the a.s. statement.
  • standard math Planarity and the FKG inequality for monotone events in the oriented lattice.
    Used to construct infinite paths from rectangle crossings and to bound crossing probabilities in Lemma 2.7.

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Pith. "Pith review of Oriented bond-site percolation in random environment and contact processes with periodic recovery." pith.science (2026). https://pith.science/paper/ATBVLGRC

@misc{pith2026250700329,
  author       = {Pith},
  title        = {Pith review of: Oriented bond-site percolation in random environment and contact processes with periodic recovery},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ATBVLGRC}},
  note         = {Machine review of arXiv:2507.00329}
}
abstract

We investigate oriented bond-site percolation on the planar lattice in which entire columns are stretched. Generalising recent results by Hil\'ario et al., we establish non-trivial percolation under a $(1+\varepsilon)$-th moment condition on the stretches and use this to prove survival of contact processes with periodic recoveries as well as in random environments.

Figures

Figures reproduced from arXiv: 2507.00329 by the authors.

Figure 1
Figure 1. Illustrations of the L 2 -lattice (left) and the OPRE on L 2 (right). Coloured vertices and edges of the same colour have the same probability to be open; for example, blue vertices are open with probability κ(ξx). by introducing environments that weaken both bonds and sites via stretches. These stretches are associ￾ated with the spatial component (i.e., the x-coordinate), and we refer to the resulting oriented bond… view at source ↗
Figure 2
Figure 2. Graphical representation of a generalised contact process with undirected edges. Vertical lines represent vertices [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Left: Realisation of the random environment given by randomly shifted periodic recovery times [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Left: Illustration of two infection events in the generalised contact process within their associated time intervals of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Left: Realisation of the random environment [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Left: Starting from the blue vertex (j, i), an infection can reach the teal vertex (j + 1, i + 1) if Ni many infection arrows exist with the correct ordering. This is equivalent to the sum of Ni i.i.d. exponential random variables of parameter λ being less than 1. This…
Figure 7
Figure 7. Figure 7: Left: We divide the process into temporal blocks of length [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: Stationary embedding of the graph in Z 2 ≥0 . Depicted is a realisation with B = 1. A key step is to identify good and bad spatial columns of the graph at different scales, for which we require the following quantities. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Depictions of the rectangle R = [t0, t1]×[a, b] (blue) together with the events LRC(R) and BTC(R). The corresponding open paths are marked in red. choose the height (time coordinate) of a rectangle exponentially in its width. To this end, we introduce another set of pa…
Figure 10
Figure 10. Figure 10: Each red line segment is an event of the form [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: Left: A site is declared open if the corresponding left-right, right-left and bottom-top crossing exist. Middle: Open [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Renormalised subgraph. Left: The state of site [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Construction of an infinite directed path based on the events [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

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