Improved Survival Results for the One-Dimensional Renewal Contact Process

Gustavo O. de Carvalho; Lucas R. de Lima

arxiv: 2605.30121 · v1 · pith:7WAQK2ZOnew · submitted 2026-05-28 · 🧮 math.PR

Improved Survival Results for the One-Dimensional Renewal Contact Process

Gustavo O. de Carvalho , Lucas R. de Lima This is my paper

Pith reviewed 2026-06-29 05:31 UTC · model grok-4.3

classification 🧮 math.PR

keywords renewal contact processcritical infection parameterarithmetic interarrival distributionoriented percolationPeierls argumentsurvival probabilityone-dimensional modelnon-Markovian process

0 comments

The pith

The one-dimensional renewal contact process has a finite critical infection rate for every non-degenerate arithmetic interarrival distribution.

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

The paper establishes that the critical infection parameter remains finite for the renewal contact process on the line when recoveries follow arbitrary non-degenerate arithmetic distributions. The same finiteness holds under a mild condition that the atomic part of the interarrival measure stays uniformly small on short intervals, which covers all non-atomic laws including singular continuous ones. The argument proceeds by deriving local renewal estimates, embedding the process into a regenerative oriented percolation model, and applying a Peierls contour argument to show positive survival probability above some finite threshold. A sympathetic reader cares because the result enlarges the set of recovery mechanisms for which the model still exhibits the same phase-transition behavior as the classical Markovian contact process.

Core claim

We prove that λ_c(μ)<+∞ for every non-degenerate arithmetic interarrival distribution. Moreover, finiteness holds whenever the atomic component of the renewal measure is uniformly small on sufficiently short intervals. This criterion applies in particular to all non-atomic interarrival distributions, including singular continuous laws. The proof combines local estimates for renewal measures with a comparison to a regenerative oriented percolation model and a Peierls-type contour argument.

What carries the argument

Comparison to a regenerative oriented percolation model, supported by local estimates on renewal measures and a Peierls-type contour argument.

If this is right

The renewal contact process exhibits a non-trivial phase transition for every non-degenerate arithmetic recovery distribution.
Survival with positive probability occurs above a finite critical infection rate for all non-atomic interarrival laws, including singular continuous ones.
The atomic-component condition on short intervals is sufficient to guarantee the result beyond the Markovian case.
Local control of renewal probabilities suffices to bound the infection spread via percolation comparison in one dimension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same local-estimate strategy might adapt to other one-dimensional interacting systems whose clocks satisfy comparable renewal regularity.
If the atomic-component condition turns out to be necessary, it would delineate a sharp boundary between distributions that permit survival and those that force extinction at all rates.
Extending the comparison technique to two dimensions would test whether the finiteness result persists when percolation arguments become more delicate.

Load-bearing premise

The local estimates for renewal measures and the comparison to a regenerative oriented percolation model remain valid under the stated conditions on the atomic component of the interarrival distribution.

What would settle it

An explicit arithmetic interarrival distribution for which the process dies out almost surely at every finite infection rate would falsify the claim that λ_c(μ) is always finite.

Figures

Figures reproduced from arXiv: 2605.30121 by Gustavo O. de Carvalho, Lucas R. de Lima.

**Figure 2.** Figure 2: Pictorial representation of the event A ↗ k,ℓ ∩ Bk+1,ℓ with M = 3. Choose λ = λ(M, d, η) sufficiently large so that for k, ℓ ∈ N, P (A ↖ k,ℓ) c = P (A ↗ k,ℓ) c < η/2. (5.2) Consider the set E as defined in (4.2) and split E = E↖ ∪ E↗, where E↖ := {(v, v′ ) ∈ V 2 : v ′ = v + (−1, 1)}, E↗ := {(v, v′ ) ∈ V 2 : v ′ = v + (1, 1)}. Define, for any edge e = (v, v′ ) ∈ E with v = (k, ℓ) ∈ V, De :=    Bk… view at source ↗

read the original abstract

The renewal contact process is a non-Markovian variant of the classical contact process in which recoveries are governed by independent renewal processes with interarrival distribution $\mu$. We establish new sufficient conditions ensuring finiteness of the critical infection parameter $\lambda_c(\mu)$ for the one-dimensional model. In particular, we prove that $\lambda_c(\mu)<+\infty$ for every non-degenerate arithmetic interarrival distribution. Moreover, finiteness holds whenever the atomic component of the renewal measure is uniformly small on sufficiently short intervals. This criterion applies in particular to all non-atomic interarrival distributions, including singular continuous laws. The proof combines local estimates for renewal measures with a comparison to a regenerative oriented percolation model and a Peierls-type contour argument.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives concrete new conditions under which the critical value stays finite for the 1D renewal contact process, including all non-degenerate arithmetic cases and all non-atomic ones.

read the letter

The headline result is that λ_c(μ) is finite whenever the interarrival distribution is arithmetic and non-degenerate, or more generally when its atomic component is uniformly small on short intervals. That last clause automatically covers every non-atomic μ, singular continuous included.

What the paper actually does is combine local renewal-measure estimates with a comparison to a regenerative oriented percolation process, then close the argument with a Peierls contour. The conditions on μ are chosen precisely so the local estimates remain uniform; once that holds, the rest follows from standard comparison and contour techniques. The arithmetic case and the uniform-small-atomic criterion look like genuine extensions of what was previously known for this model.

The main limitation is that everything stays in one dimension and the argument yields no quantitative bounds on λ_c. The proof outline does not appear to hide circularity or extra uniformity assumptions beyond what is stated, but the local estimates themselves are the load-bearing step and would need careful checking in the full text.

This is a narrow but clean incremental result aimed at people who already work on contact processes and renewal-driven particle systems. It is worth sending to a serious referee; the claim is specific enough that referees can verify the estimates without needing to resolve the general higher-dimensional case.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that the critical infection rate λ_c(μ) is finite for the one-dimensional renewal contact process whenever the interarrival distribution μ is non-degenerate and arithmetic, and more generally whenever the atomic component of the associated renewal measure is uniformly small on sufficiently short intervals. This criterion covers all non-atomic distributions, including singular continuous laws. The argument proceeds by establishing local estimates for the renewal measure, comparing the process to a regenerative oriented percolation model, and controlling the latter via a Peierls-type contour argument.

Significance. If the local estimates and percolation comparison hold under the stated hypotheses on the atomic component, the result substantially enlarges the class of interarrival distributions for which survival occurs at small λ. In particular, it settles the finiteness question for all arithmetic μ and for singular continuous μ, cases that had remained open. The proof strategy is standard in the field but requires uniform control on the renewal measure that the atomic-smallness condition supplies directly.

minor comments (3)

§1, paragraph following the statement of Theorem 1.1: the phrase 'non-degenerate arithmetic' is used without an explicit definition; a one-sentence clarification of what 'non-degenerate' means for the support of μ would prevent ambiguity with the later atomic-component hypothesis.
§3.2, display (3.4): the constant C appearing in the local renewal estimate is stated to depend only on the length of the short interval, but the dependence on the total mass of the atomic component is not recorded; adding this dependence would make the uniformity claim fully explicit.
Figure 2: the caption refers to 'typical contour configurations' but does not indicate the scale or the value of λ used in the simulation; a brief note on the parameters would improve reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and detailed assessment of our manuscript, including the recognition that the result settles the finiteness question for arithmetic and singular continuous interarrival distributions. The recommendation for minor revision is noted. No major comments were listed in the report.

Circularity Check

0 steps flagged

No circularity: direct proof via estimates and comparison

full rationale

The derivation relies on local renewal-measure estimates, comparison to a regenerative oriented percolation process, and a Peierls contour argument under explicit hypotheses on the atomic component of μ. These are independent mathematical steps that do not reduce to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations; the sufficient condition on atomic mass is precisely the hypothesis that makes the estimates uniform, with no internal reduction to the target result λ_c(μ)<∞. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard background facts about renewal processes and oriented percolation; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Standard properties of renewal measures and the contact process on Z
Invoked implicitly when defining the model and applying comparison arguments.

pith-pipeline@v0.9.1-grok · 5648 in / 1166 out tokens · 21969 ms · 2026-06-29T05:31:02.235188+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 10 canonical work pages

[1]

Durrett.Ten lectures on particle systems, volume 1608 ofLecture Notes in Math

R. Durrett.Ten lectures on particle systems, volume 1608 ofLecture Notes in Math. Springer, Berlin, 1995. ISBN 3-540-60015-9. doi: 10.1007/BFb0095747. URLhttps: //doi.org/10.1007/BFb0095747

work page doi:10.1007/bfb0095747 1995
[2]

Feller.An introduction to probability theory and its applications, Volume 2, volume 2

W. Feller.An introduction to probability theory and its applications, Volume 2, volume 2. John Wiley & Sons, 1991

1991
[3]

L. R. Fontes, T. S. Mountford, and M. E. Vares. Contact process under renewals II. Stochastic Process. Appl., 130(2):1103–1118, 2020. ISSN 0304-4149,1879-209X. doi: 10.1016/j.spa.2019.04.008. URLhttps://doi.org/10.1016/j.spa.2019.04.008

work page doi:10.1016/j.spa.2019.04.008 2020
[4]

L. R. Fontes, T. S. Mountford, D. Ungaretti, and M. E. Vares. Renewal contact processes: phase transition and survival.Stochastic Process. Appl., 161:102–136, 2023. ISSN 0304-

2023
[5]

URLhttps://doi.org/10.1016/j.spa.2023

doi: 10.1016/j.spa.2023.03.005. URLhttps://doi.org/10.1016/j.spa.2023. 03.005

work page doi:10.1016/j.spa.2023.03.005 2023
[6]

L. R. G. Fontes, D. H. U. Marchetti, T. S. Mountford, and M. E. Vares. Contact process under renewals I.Stochastic Process. Appl., 129(8):2903–2911, 2019. ISSN 0304-4149. doi: 10.1016/j.spa.2018.08.007. URLhttps://doi.org/10.1016/j.spa.2018.08.007

work page doi:10.1016/j.spa.2018.08.007 2019
[7]

Grimmett.Percolation, volume 321 ofGrundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

G. Grimmett.Percolation, volume 321 ofGrundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1999. ISBN 3-540-64902-6. doi: 10.1007/978-3-662-03981-6. URL https://doi.org/10.1007/978-3-662-03981-6

work page doi:10.1007/978-3-662-03981-6 1999
[8]

T. E. Harris. Contact interactions on a lattice.Ann. Probability, 2:969–988, 1974. ISSN 0091-1798. doi: 10.1214/aop/1176996493. URLhttps://doi.org/10.1214/aop/ 1176996493

work page doi:10.1214/aop/1176996493 1974
[9]

T. E. Harris. Additive set-valued Markov processes and graphical methods.Ann. Probability, 6(3):355–378, 1978. ISSN 0091-1798. doi: 10.1214/aop/1176995523. URL https://doi.org/10.1214/aop/1176995523. 17

work page doi:10.1214/aop/1176995523 1978
[10]

Hil´ ario, D

M. Hil´ ario, D. Ungaretti, D. Valesin, and M. E. Vares. Results on the contact process with dynamic edges or under renewals.Electron. J. Probab., 27:Paper No. 91, 31, 2022. doi: 10.1214/22-ejp811. URLhttps://doi.org/10.1214/22-ejp811

work page doi:10.1214/22-ejp811 2022
[11]

T. M. Liggett.Interacting particle systems. Classics in Mathematics. Springer-Verlag, Berlin, 2005. ISBN 3-540-22617-6. doi: 10.1007/b138374. URLhttps://doi.org/10. 1007/b138374. Reprint of the 1985 original

work page doi:10.1007/b138374 2005
[12]

Santos and M

R. Santos and M. E. Vares. Survival of one dimensional renewal contact process.ALEA Lat. Am. J. Probab. Math. Stat., 21(2):1823–1833, 2024. doi: 10.30757/alea.v21-68. URL https://doi.org/10.30757/alea.v21-68. 18

work page doi:10.30757/alea.v21-68 2024

[1] [1]

Durrett.Ten lectures on particle systems, volume 1608 ofLecture Notes in Math

R. Durrett.Ten lectures on particle systems, volume 1608 ofLecture Notes in Math. Springer, Berlin, 1995. ISBN 3-540-60015-9. doi: 10.1007/BFb0095747. URLhttps: //doi.org/10.1007/BFb0095747

work page doi:10.1007/bfb0095747 1995

[2] [2]

Feller.An introduction to probability theory and its applications, Volume 2, volume 2

W. Feller.An introduction to probability theory and its applications, Volume 2, volume 2. John Wiley & Sons, 1991

1991

[3] [3]

L. R. Fontes, T. S. Mountford, and M. E. Vares. Contact process under renewals II. Stochastic Process. Appl., 130(2):1103–1118, 2020. ISSN 0304-4149,1879-209X. doi: 10.1016/j.spa.2019.04.008. URLhttps://doi.org/10.1016/j.spa.2019.04.008

work page doi:10.1016/j.spa.2019.04.008 2020

[4] [4]

L. R. Fontes, T. S. Mountford, D. Ungaretti, and M. E. Vares. Renewal contact processes: phase transition and survival.Stochastic Process. Appl., 161:102–136, 2023. ISSN 0304-

2023

[5] [5]

URLhttps://doi.org/10.1016/j.spa.2023

doi: 10.1016/j.spa.2023.03.005. URLhttps://doi.org/10.1016/j.spa.2023. 03.005

work page doi:10.1016/j.spa.2023.03.005 2023

[6] [6]

L. R. G. Fontes, D. H. U. Marchetti, T. S. Mountford, and M. E. Vares. Contact process under renewals I.Stochastic Process. Appl., 129(8):2903–2911, 2019. ISSN 0304-4149. doi: 10.1016/j.spa.2018.08.007. URLhttps://doi.org/10.1016/j.spa.2018.08.007

work page doi:10.1016/j.spa.2018.08.007 2019

[7] [7]

Grimmett.Percolation, volume 321 ofGrundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

G. Grimmett.Percolation, volume 321 ofGrundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1999. ISBN 3-540-64902-6. doi: 10.1007/978-3-662-03981-6. URL https://doi.org/10.1007/978-3-662-03981-6

work page doi:10.1007/978-3-662-03981-6 1999

[8] [8]

T. E. Harris. Contact interactions on a lattice.Ann. Probability, 2:969–988, 1974. ISSN 0091-1798. doi: 10.1214/aop/1176996493. URLhttps://doi.org/10.1214/aop/ 1176996493

work page doi:10.1214/aop/1176996493 1974

[9] [9]

T. E. Harris. Additive set-valued Markov processes and graphical methods.Ann. Probability, 6(3):355–378, 1978. ISSN 0091-1798. doi: 10.1214/aop/1176995523. URL https://doi.org/10.1214/aop/1176995523. 17

work page doi:10.1214/aop/1176995523 1978

[10] [10]

Hil´ ario, D

M. Hil´ ario, D. Ungaretti, D. Valesin, and M. E. Vares. Results on the contact process with dynamic edges or under renewals.Electron. J. Probab., 27:Paper No. 91, 31, 2022. doi: 10.1214/22-ejp811. URLhttps://doi.org/10.1214/22-ejp811

work page doi:10.1214/22-ejp811 2022

[11] [11]

T. M. Liggett.Interacting particle systems. Classics in Mathematics. Springer-Verlag, Berlin, 2005. ISBN 3-540-22617-6. doi: 10.1007/b138374. URLhttps://doi.org/10. 1007/b138374. Reprint of the 1985 original

work page doi:10.1007/b138374 2005

[12] [12]

Santos and M

R. Santos and M. E. Vares. Survival of one dimensional renewal contact process.ALEA Lat. Am. J. Probab. Math. Stat., 21(2):1823–1833, 2024. doi: 10.30757/alea.v21-68. URL https://doi.org/10.30757/alea.v21-68. 18

work page doi:10.30757/alea.v21-68 2024