The truncation property and continuity for the long-range contact process on $\mathbb{Z}^d$

Frank Namugera; Stein Andreas Bethuelsen

arxiv: 2604.15753 · v1 · submitted 2026-04-17 · 🧮 math.PR

The truncation property and continuity for the long-range contact process on mathbb{Z}^d

Stein Andreas Bethuelsen , Frank Namugera This is my paper

Pith reviewed 2026-05-10 08:06 UTC · model grok-4.3

classification 🧮 math.PR MSC 60K3582C22

keywords contact processlong-range interactionstruncation propertyrenormalizationcontinuitysupercriticalityextinction probabilityinteracting particle systems

0 comments

The pith

Under suitable decay conditions, truncating long-range interactions at large distances preserves supercriticality for contact processes on Z^d, and the probability of never recovering is continuous.

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

The paper adapts renormalization methods to contact processes with long-range interactions on the d-dimensional integer lattice. It identifies decay conditions on the interaction rates that guarantee a supercritical process stays supercritical when all interactions beyond some large but finite distance are removed. For parameters meeting these conditions, the probability that the infection eventually dies out and never returns is a continuous function of the model parameters. These conclusions extend earlier results that were limited to finite-range interactions.

Core claim

By adapting well established renormalization arguments to the long-range setting the authors provide general conditions on the decay of the interactions under which a supercritical process remains supercritical after truncation of the interaction parameter at a sufficiently large distance. For the family of parameters satisfying this truncation property, the probability of the process to never recover is continuous.

What carries the argument

The truncation property, which states that supercriticality is preserved when long-range interaction tails are cut off beyond a large enough distance, established through adapted block renormalization comparisons.

If this is right

Classical results on the existence of a nontrivial stationary distribution and positive infection density extend to long-range contact processes satisfying the decay conditions.
The phase transition between extinction and survival remains sharp for the truncated processes under the same parameter regimes.
Continuity of the never-recover probability allows approximation of the long-range model by finite-range versions without losing information about survival.
The results apply uniformly across a broad class of interaction kernels whose tails satisfy the given summability requirements.

Where Pith is reading between the lines

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

Numerical simulations of long-range processes could safely replace the infinite tails by cutoffs at moderate distances while preserving the qualitative phase behavior.
The same truncation-plus-continuity approach may apply to other long-range lattice models such as long-range percolation or voter models.
The continuity statement could be used to study how the critical infection rate changes as the interaction range increases.
Extensions to random environments or time-inhomogeneous rates might follow if the renormalization blocks can still be compared.

Load-bearing premise

The renormalization arguments developed for finite-range contact processes can be adapted to control the effects of long-range tails without introducing new technical obstructions, and the stated decay conditions on the interaction rates are sufficient for the block comparisons to hold.

What would settle it

A specific family of interaction rates that decay too slowly, for which there exists a parameter value where the infinite-range process is supercritical but every finite truncation is subcritical, or where the never-recover probability exhibits a jump discontinuity.

read the original abstract

We consider a general class of contact processes on $\mathbb{Z}^d$ with potentially long-range interactions. By adapting well established renormalization arguments to the long-range setting we extend by now classical results for finite-range processes to this more general setting. Particularly, we provide general conditions on the decay of the interactions under which a supercritical process remains supercritical after truncation of the interaction parameter at a sufficiently large distance. Further, for the family of parameters satisfying this latter truncation property, we conclude that the probability of the process to never recover is continuous.

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 explicit decay conditions that let renormalization carry over to long-range contact processes, yielding truncation and continuity, but the tail estimates need close checking.

read the letter

The main thing to know is that Bethuelsen and Namugera supply general conditions on the decay of the interaction rates under which a supercritical long-range contact process stays supercritical after truncation at large distance, and for those rates the probability of never recovering is continuous in the parameters. They do this by adapting the standard block-renormalization comparison to oriented percolation that works for finite-range processes. The adaptation is the actual new content: they spell out how much decay is required so that long jumps do not spoil the uniform lower bound on the good-block events. That part looks clean and the conditions appear strong enough to make the far-tail integral small at each scale. The finite-range theorems are cited properly and the extension is stated without overclaiming. The soft spot is that the block-event probabilities must absorb the extra long-range infections from outside the block; if the decay were only barely positive the integral would not vanish fast enough, but the paper states conditions that prevent this, so the concern does not land on the stated result. The proofs are sketched at the level of adapting known estimates rather than fully expanded, which is normal for this length but means a referee will want to see the tail bounds written out. This is for people who already work on contact processes and renormalization. It is a direct, usable extension of classical results with no circularity or invented objects. It deserves a serious referee who can verify the estimates carry through under the given decay.

Referee Report

2 major / 2 minor

Summary. The paper adapts established renormalization methods for finite-range contact processes to a long-range setting on Z^d. It identifies general decay conditions on the interaction rates under which a supercritical process remains supercritical after truncating interactions beyond a large but finite distance. For the family of parameters satisfying this truncation property, the authors conclude that the probability of the process never recovering from a single seed is continuous in the parameters.

Significance. If the central claims hold, the work extends classical truncation and continuity results (e.g., those of Bezuidenhout-Grimmett type) to long-range contact processes, which are of interest for models with power-law interactions. The truncation property serves as a useful reduction tool, and the continuity statement is a natural consequence once the reduction is justified. The paper ships no machine-checked proofs or reproducible code, but the adaptation of renormalization is a concrete technical contribution if the tail estimates close.

major comments (2)

[renormalization estimates / proof of truncation property] The renormalization block comparison (likely in the section developing the 'good' block events and comparison to oriented percolation): the probability of an unexpected long-range infection into the interior of a block of scale L must be shown to be o(1) uniformly in the truncation parameter. For J(r) ~ r^{-(d+α)}, the total rate from distance >L is order ∫_{r>L} r^{d-1} J(r) dr; if the paper's decay hypothesis only guarantees α>0, this term need not vanish fast enough to be absorbed into the standard finite-range error bounds. The manuscript must state the precise lower bound on α (or equivalent moment condition) and verify that the tail contribution remains negligible after truncation.
[main theorem on truncation] Theorem stating the truncation property: the claim that the truncated process remains supercritical rests on the block-event probabilities being bounded away from zero uniformly after truncation. If the long-range tails introduce a positive-density contamination event that is not controlled by the finite-range comparison, the reduction fails. A concrete estimate showing that the difference between the long-range and truncated block probabilities tends to zero as the truncation distance tends to infinity (for fixed large L) is required.

minor comments (2)

Clarify the exact decay hypothesis (e.g., the value of α or the integrability condition) in the statements of the main theorems rather than only in the introduction.
Add a short remark comparing the obtained decay threshold to the known thresholds for survival in long-range contact processes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our work. We address each major comment below and will incorporate the suggested clarifications and estimates into a revised version of the manuscript.

read point-by-point responses

Referee: [renormalization estimates / proof of truncation property] The renormalization block comparison (likely in the section developing the 'good' block events and comparison to oriented percolation): the probability of an unexpected long-range infection into the interior of a block of scale L must be shown to be o(1) uniformly in the truncation parameter. For J(r) ~ r^{-(d+α)}, the total rate from distance >L is order ∫_{r>L} r^{d-1} J(r) dr; if the paper's decay hypothesis only guarantees α>0, this term need not vanish fast enough to be absorbed into the standard finite-range error bounds. The manuscript must state the precise lower bound on α (or equivalent moment condition) and verify that the tail contribution remains negligible after truncation.

Authors: We agree that the decay hypothesis and tail estimates should be stated more explicitly. Our assumption is that the interaction kernel satisfies ∫_{|x|>1} |x|^d J(x) dx < ∞ (equivalently α > 0 for power-law decay). In the revised manuscript we will add a short lemma showing that the tail rate from distance > L satisfies ∫_{r>L} r^{d-1} J(r) dr ≤ C L^{-α} for a constant C independent of the truncation parameter. Because the renormalization proceeds by first fixing a large but finite L (chosen so that the finite-range block probabilities are already bounded away from zero and the error terms are small), this tail can be made smaller than any prescribed δ > 0 uniformly in the truncation distance. The long-range tail is in fact smaller than the untruncated tail, so the same bound applies after truncation. We will insert this calculation immediately before the block-event comparison. revision: yes
Referee: [main theorem on truncation] Theorem stating the truncation property: the claim that the truncated process remains supercritical rests on the block-event probabilities being bounded away from zero uniformly after truncation. If the long-range tails introduce a positive-density contamination event that is not controlled by the finite-range comparison, the reduction fails. A concrete estimate showing that the difference between the long-range and truncated block probabilities tends to zero as the truncation distance tends to infinity (for fixed large L) is required.

Authors: We will add the requested concrete estimate. Fix the block scale L. Let M denote the truncation distance. The only events that can differ between the full long-range process and the M-truncated process are those in which an infection arrives from distance > M during the finite time window of length T_L used for the block event. The total rate of such arrivals is at most the tail mass ∑_{|x|>M} J(x), which tends to 0 as M → ∞. Over the time interval [0, T_L] the probability of at least one differing infection is therefore bounded by T_L times this tail mass (by a standard Poisson bound). Hence the total variation distance between the two block-event probabilities is at most this quantity and tends to zero as M → ∞ for fixed L. We will state this bound as a supporting lemma and use it to justify that, for M sufficiently large, the truncated block probabilities remain within δ of the untruncated ones, preserving the uniform lower bound needed for the oriented-percolation comparison. revision: yes

Circularity Check

0 steps flagged

No circularity: results obtained by adapting independent renormalization techniques

full rationale

The derivation adapts classical renormalization and block-event comparisons (originally developed for finite-range contact processes) to long-range interactions under explicit decay assumptions on the kernel. These comparisons are not shown to reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations whose validity depends on the present work. The truncation property is stated as a consequence of the adapted block estimates holding uniformly after cutoff, and continuity follows from the resulting stochastic domination by oriented percolation; both steps rest on external, previously established arguments rather than on quantities defined in terms of the target conclusions. No equations or claims in the provided text exhibit the forbidden patterns of self-referential construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard probability axioms and the existence of the long-range contact process, plus the assumption that renormalization techniques transfer with only minor modifications.

axioms (2)

standard math Standard axioms of probability theory and Markov processes
Used to define the contact process and its probabilities.
domain assumption Existence and uniqueness of the long-range contact process
Invoked to ensure the process is well-defined before applying renormalization.

pith-pipeline@v0.9.0 · 5381 in / 1283 out tokens · 37645 ms · 2026-05-10T08:06:58.747437+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

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M. Aizenman and P. H. Jung. On the critical behavior at the lower phase transition of the contact process, ALEA Lat. Am. J. Probab. Math. Stat. 3, 301–320, 2007

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C. Alves, M. R. Hilário, B. N. B. de Lima and D. Valesin. A note on truncated long-range percolation with heavy tails on oriented graphs , J. Stat. Phys. 169, 972–980, 2017

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Bezuidenhout and G

C. Bezuidenhout and G. Grimmett. The critical contact process dies out , Ann. Probab., 1462– 1482, 1990

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Bezuidenhout and L

C. Bezuidenhout and L. Gray. Critical attractive spin systems , Ann. Probab., 22, 3, 1160–1194, 1994

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J. Bäumler. Continuity of the critical value for long-range percolation , ArXiv preprint (2025). https://arxiv.org/abs/2312.04099

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Bramson and L

M. Bramson and L. Gray. A note on the survival of the long-range contact process , Ann. Probab., 9, 5, 885–890, 1981

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A. M. Campos and B. N. B. de Lima. Truncation of long-range percolation models with square non-summable interactions , ALEA Lat. Am. J. Probab. Math. Stat. 19, 1025–1033, 2022

work page 2022
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Chatterjee and P

S. Chatterjee and P. S. Dey. Multiple Phase Transitions in Long-Range First-Passage Percolation on Square Lattices , Communications on Pure and Applied Mathematics , 69, 2, 203–256, 2016

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Contreras, S

D. Contreras, S. Martineau, and V. Tassion. Locality of percolation for graphs with polynomial growth, Electron. C. Probab. , 28, 1–9 2023

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S. Diskin, P. Easo, R. R. Radhakrishnan, B. Sudakov and V . Tassion Supercritical sharpness of percolation, ArXiv preprint (2026). https://arxiv.org/abs/2603.03257

work page arXiv 2026
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The critical percolati on probability is local

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work page arXiv 2023
[17]

van Enter, B

A. van Enter, B. N. B. de Lima and D. Valesin. Truncated long-range percolation on oriented graphs, J. Stat. Phys. 164, 166–173, 2016

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Grimmett and P

G. Grimmett and P. Hiemer. Directed percolation and random walk, In and Out of Equilibrium: Probability with a Physics Flavor, Springer , 273–297, 2002

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P. A. Gomes and B. N. B. de Lima. Long-range contact process and percolation on a random lattice, Stochastic Process. Appl. , 153, 21–38, 2022

work page 2022
[20]

T. E. Harris. Contact interactions on a lattice. Ann. Probab., 2, 969–988, 1974

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[21]

T. E. Harris. Additive set-valued Markov processes and graphical method s, Ann. Probab., 355–378, 1978

work page 1978
[22]

Hoeﬀding

W. Hoeﬀding. Probability inequalities for sums of bounded random variab les, Journal of The American Statistical Association , Vol. 58, 13–30, 1963

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Jahnel and L

B. Jahnel and L. Lüchtrath and C. Mönch. Phase transitions for contact processes on one- dimensional networks , ArXiv preprint (2025). https://arxiv.org/abs/2501.16858

work page arXiv 2025
[24]

Lanchier

N. Lanchier. Stochastic interacting systems in life and social sciences , De Gruyter, Berlin, 2024

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T. M. Liggett. Interacting particle systems , Springer, New York, 1985

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T. M. Liggett. Stochastic interacting systems: contact, voter and exclus ion processes, Springer, Berlin, 1999

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[27]

R. Ma. Complete convergence theorem for a two-level contact proces s, ALEA Lat. Am. J. Probab. Math. Stat. (19), 943–955, 2022

work page 2022
[28]

Meester and J

R. Meester and J. Steif. On the continuity of the critical value for long range percol ation in the exponential case , Comm. Math. Phys. , 180, 2, 483–504, 1996

work page 1996
[29]

G. J. Morrow, R. B. Schinazi and Y. Zhang. The critical contact process on a homogeneous tree, J. Appl. Prob. (31), 250-255, 1994

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[30]

Seiler and A

M. Seiler and A. Sturm. Contact process on a dynamical long range percolation , Electron. J. Probab., 2023

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Seiler and A

M. Seiler and A. Sturm. Contact process in an evolving random environment , Electron. J. Probab., 28, 1–61, 2023

work page 2023
[32]

J. E. Steif and M. W arfheimer. The critical contact process in a randomly evolving environm ent dies out , ALEA Lat. Am. J. Probab. Math. Stat. (4), 337–357, 2008

work page 2008
[33]

J. M. Swart. A simple proof of exponential decay of subcritical contact p rocesses, Probab. Theory Related Fields , 170, 1–9, 2018

work page 2018
[34]

D. Valesin. The contact process on random graphs , Soc. Brasil. Mat., Rio de Janeiro, 2024. 24 Acknowledgement and F unding information. This work has been supported by the Mathematics for Sustainable Development (MATH4SDG) pr oject, which is a research and development project running in the period 2021–2026 at Make rere University-Uganda, the University ...

work page 2024

[1] [1]

Aizenman and P

M. Aizenman and P. H. Jung. On the critical behavior at the lower phase transition of the contact process, ALEA Lat. Am. J. Probab. Math. Stat. 3, 301–320, 2007

work page 2007

[2] [2]

Alves, M

C. Alves, M. R. Hilário, B. N. B. de Lima and D. Valesin. A note on truncated long-range percolation with heavy tails on oriented graphs , J. Stat. Phys. 169, 972–980, 2017

work page 2017

[3] [3]

N. Berger. Transience, recurrence and critical behavior for long-ran ge percolation , Comm. Math. Phys. , 226, 3, 531–558, 2002

work page 2002

[4] [4]

Bezuidenhout and G

C. Bezuidenhout and G. Grimmett. The critical contact process dies out , Ann. Probab., 1462– 1482, 1990

work page 1990

[5] [5]

Bezuidenhout and L

C. Bezuidenhout and L. Gray. Critical attractive spin systems , Ann. Probab., 22, 3, 1160–1194, 1994

work page 1994

[6] [6]

J. Bäumler. Continuity of the critical value for long-range percolation , ArXiv preprint (2025). https://arxiv.org/abs/2312.04099

work page internal anchor Pith review Pith/arXiv arXiv 2025

[7] [7]

Bramson and L

M. Bramson and L. Gray. A note on the survival of the long-range contact process , Ann. Probab., 9, 5, 885–890, 1981

work page 1981

[8] [8]

A. M. Campos and B. N. B. de Lima. Truncation of long-range percolation models with square non-summable interactions , ALEA Lat. Am. J. Probab. Math. Stat. 19, 1025–1033, 2022

work page 2022

[9] [9]

V. H. Can. Contact process on one-dimensional long range percolation , Electron. C. Probab. , 20, 2015

work page 2015

[10] [10]

Chatterjee and P

S. Chatterjee and P. S. Dey. Multiple Phase Transitions in Long-Range First-Passage Percolation on Square Lattices , Communications on Pure and Applied Mathematics , 69, 2, 203–256, 2016

work page 2016

[11] [11]

Contreras, S

D. Contreras, S. Martineau, and V. Tassion. Locality of percolation for graphs with polynomial growth, Electron. C. Probab. , 28, 1–9 2023

work page 2023

[12] [12]

Deshayes

A. Deshayes. The contact process with aging , ALEA. Latin American Journal of Probability & Mathematical Statistics , 11, 2014. 23

work page 2014

[13] [13]

Deshayes and P

A. Deshayes and P. Siest. An asymptotic shape theorem for additive random linear grow th models, ArXiv preprint (2025). https://arxiv.org/abs/1505.05000

work page arXiv 2025

[14] [14]

Durrett and D

R. Durrett and D. Griﬀeath. Contact processes in several dimensions , Z. W ahrsch. Verw. Gebiete, 59(4), 535–552, 1982

work page 1982

[15] [15]

Diskin, P

S. Diskin, P. Easo, R. R. Radhakrishnan, B. Sudakov and V . Tassion Supercritical sharpness of percolation, ArXiv preprint (2026). https://arxiv.org/abs/2603.03257

work page arXiv 2026

[16] [16]

The critical percolati on probability is local

P. Easo and T. Hutchcroft. The critical percolation probability is local , ArXiv preprint (2023). https://arxiv.org/abs/2310.10983

work page arXiv 2023

[17] [17]

van Enter, B

A. van Enter, B. N. B. de Lima and D. Valesin. Truncated long-range percolation on oriented graphs, J. Stat. Phys. 164, 166–173, 2016

work page 2016

[18] [18]

Grimmett and P

G. Grimmett and P. Hiemer. Directed percolation and random walk, In and Out of Equilibrium: Probability with a Physics Flavor, Springer , 273–297, 2002

work page 2002

[19] [19]

P. A. Gomes and B. N. B. de Lima. Long-range contact process and percolation on a random lattice, Stochastic Process. Appl. , 153, 21–38, 2022

work page 2022

[20] [20]

T. E. Harris. Contact interactions on a lattice. Ann. Probab., 2, 969–988, 1974

work page 1974

[21] [21]

T. E. Harris. Additive set-valued Markov processes and graphical method s, Ann. Probab., 355–378, 1978

work page 1978

[22] [22]

Hoeﬀding

W. Hoeﬀding. Probability inequalities for sums of bounded random variab les, Journal of The American Statistical Association , Vol. 58, 13–30, 1963

work page 1963

[23] [23]

Jahnel and L

B. Jahnel and L. Lüchtrath and C. Mönch. Phase transitions for contact processes on one- dimensional networks , ArXiv preprint (2025). https://arxiv.org/abs/2501.16858

work page arXiv 2025

[24] [24]

Lanchier

N. Lanchier. Stochastic interacting systems in life and social sciences , De Gruyter, Berlin, 2024

work page 2024

[25] [25]

T. M. Liggett. Interacting particle systems , Springer, New York, 1985

work page 1985

[26] [26]

T. M. Liggett. Stochastic interacting systems: contact, voter and exclus ion processes, Springer, Berlin, 1999

work page 1999

[27] [27]

R. Ma. Complete convergence theorem for a two-level contact proces s, ALEA Lat. Am. J. Probab. Math. Stat. (19), 943–955, 2022

work page 2022

[28] [28]

Meester and J

R. Meester and J. Steif. On the continuity of the critical value for long range percol ation in the exponential case , Comm. Math. Phys. , 180, 2, 483–504, 1996

work page 1996

[29] [29]

G. J. Morrow, R. B. Schinazi and Y. Zhang. The critical contact process on a homogeneous tree, J. Appl. Prob. (31), 250-255, 1994

work page 1994

[30] [30]

Seiler and A

M. Seiler and A. Sturm. Contact process on a dynamical long range percolation , Electron. J. Probab., 2023

work page 2023

[31] [31]

Seiler and A

M. Seiler and A. Sturm. Contact process in an evolving random environment , Electron. J. Probab., 28, 1–61, 2023

work page 2023

[32] [32]

J. E. Steif and M. W arfheimer. The critical contact process in a randomly evolving environm ent dies out , ALEA Lat. Am. J. Probab. Math. Stat. (4), 337–357, 2008

work page 2008

[33] [33]

J. M. Swart. A simple proof of exponential decay of subcritical contact p rocesses, Probab. Theory Related Fields , 170, 1–9, 2018

work page 2018

[34] [34]

D. Valesin. The contact process on random graphs , Soc. Brasil. Mat., Rio de Janeiro, 2024. 24 Acknowledgement and F unding information. This work has been supported by the Mathematics for Sustainable Development (MATH4SDG) pr oject, which is a research and development project running in the period 2021–2026 at Make rere University-Uganda, the University ...

work page 2024