Pith. sign in

REVIEW 4 major objections 4 minor 2 cited by

Phase transitions for contact processes on one-dimensional networks

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A finiteness condition on edge crossings across each cut forces a positive epidemic threshold on stationary one-dimensional random graphs.

desk verdict Strong results and a promising new mechanism, but the central RWRE step in Theorem 2.2 has a real gap that needs fixing before the proof holds. read the letter →

arxiv 2501.16858 v2 pith:NQ5DTK5S submitted 2025-01-28 math.PR

Benedikt Jahnel , Lukas Lüchtrath , Christian Mönch This is my paper · ORCID
classification math.PR MSC 60K3505C8291D30
keywords contactprocessphasetransitionlong-rangepercolationGilbertgraphrandomgeometricscale-freenetworkSISepidemicswalkinenvironment
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 contact process is a basic model of infection spread: infected vertices pass the infection to neighbours at rate $\lambda$ and recover at rate $1$. This paper asks when a random infinite network built on the integers has a genuine extinction phase, meaning the infection dies out for small $\lambda$ and can survive only for large $\lambda$. The main theorem says this happens whenever the network is stationary and ergodic, has finite expected degree, and almost surely only finitely many long edges cross each cut between the left and right half-lines. That condition is mild enough to cover heavy-tailed spatial networks where earlier techniques, which required exponentially decaying degrees, failed. In particular, the result settles the long-range percolation threshold $\delta>2$ and gives a sharp integrability condition for geographic Gilbert graphs.

What carries the argument

The central object is the cut-point block decomposition of $\mathbb{Z}$. A link is a $K$-cut point when at most $K$ edges of the graph connect its left side to its right side; the almost-sure finiteness of $e$ makes cut points occur with positive density, and ergodicity makes the blocks between consecutive cut points a stationary sequence of finite random graphs with finite expected length and finite expected edge count. The proof then couples the contact process on these blocks to a nearest-neighbour random walk in a random environment on $\mathbb{N}\cup\{0\}$, where the environment is the random block structure. The key identity is the recurrence criterion $\int \log((1-\omega)/\omega)\,dQ \ge 0$ for the random walk's jump probabilities $\omega$; when $\lambda$ is small the block environment makes this integral positive, so the walk recurs, and recurrence of the block process is what converts survival into extinction.

What would settle it

To test the theorem, simulate the contact process on an augmented Gilbert graph on $\mathbb{Z}$ built from i.i.d. radii with finite mean, at a small infection rate such as $\lambda=0.1$, on finite windows of growing length; the theorem predicts the infection dies out. If the survival probability stays bounded away from zero as the window grows, the asserted positive critical rate is false. A complementary test is to construct a stationary, ergodic, sparse graph on $\mathbb{Z}$ with almost surely finite edge crossings per link and measure whether $\lambda_c$ is positive; the theorem says every such graph has $\lambda_c>0$.

Watch

Extended reading notes

Core claim

The paper's main claim is Theorem 2.2: for a random graph on $\mathbb{Z}$ satisfying stationarity and ergodicity, finite expected root degree, and $\mathbb{P}(e<\infty)=1$, where $e$ is the number of edges crossing the link between $-1$ and $0$, the critical infection rate satisfies $\lambda_c(G)\in(0,\infty)$ almost surely. The condition $\mathbb{P}(e<\infty)=1$ means cut points exist with positive density: links crossed by only one or a few edges split the line into finite blocks. The proof shows that the rightmost infected particle, viewed on the block chain, is controlled by a random walk in a random environment, and a classical recurrence criterion makes that walk recurrent for all sufficiently small $\lambda$. Recurrence forces the infection to return to a single point infinitely often, which is exactly extinction. The same mechanism yields a subcritical phase for long-range percolation with connection probabilities of order $|x-y|^{-\delta}$ for every $\delta>2$, for augmented Gilbert graphs if and only if the radius distribution has finite mean, and it shows that in dimensions $d\ge 2$ heavy power-law tails with exponent $\tau\le d+1$ destroy the subcritical phase.

Load-bearing premise

The entire argument depends on almost every link of the integer line being crossed by only finitely many long edges; if infinitely many long edges cross every link almost surely, cut points have probability zero, the block decomposition never starts, and the random-walk coupling cannot be built.

Editorial extensions

If this is right

  • Long-range percolation on $\mathbb{Z}$ with connection probability of order $|x-y|^{-\delta}$ has a positive epidemic threshold for every $\delta>2$, verifying the standing conjecture and improving the older bound $\delta^{*}\le 102$.
  • For augmented Gilbert graphs on the line with i.i.d. radii, the subcritical phase exists if and only if the typical radius has finite mean; infinite mean makes the graph fail to be locally finite and forces $\lambda_c=0$.
  • In dimensions $d\ge 2$, spatial Boolean models with power-law degree exponent $\tau\le d+1$ have $\lambda_c=0$, so without at least a finite $d$-th degree moment there is no extinction phase.
  • Heavy-tailed degree distributions do not by themselves rule out a non-trivial epidemic phase when long edges are geometrically scarce.

Reading between the lines

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

  • Editorial extension: the recurrence criterion in the proof is quantitative, so the same block decomposition can yield explicit lower bounds for $\lambda_c$ from the laws of block length and block edge count, which the paper does not work out.
  • Editorial extension: the theorem does not decide what happens when $e=\infty$ almost surely; we infer that some of those graphs may still have $\lambda_c>0$, and the critical scale-invariant long-range percolation model is a natural place to test this numerically.
  • Editorial extension: viewed as an epidemic statement, the result suggests that on one-dimensional heavy-tailed networks the way to create a true epidemic threshold is to suppress very long edges rather than to reduce mean degree; this policy reading is not in the paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the contact process on stationary ergodic random graphs with vertex set Z, augmented by long-range edges. The main result, Theorem 2.2, states that if the random graph satisfies (A.1) stationarity and ergodicity, (A.2) finite expected root degree, and the number e of edges crossing a single link is almost surely finite, then the critical infection rate satisfies λ_c∈(0,∞) almost surely. The proof decomposes the graph into blocks between cut points, constructs a dominating contact process whose rightmost infection is tracked, and claims that the block-index jump chain is a random walk in a stationary ergodic random environment; Ledrappier's criterion is then invoked to show recurrence for small λ. Applications are given to long-range percolation (improving Can's bound to δ>2), augmented Gilbert graphs, weight-dependent random connection models, and a higher-dimensional no-phase-transition result for heavy-tailed Boolean models.

Significance. If correct, Theorem 2.2 is a broad and simple sufficient condition for nontrivial phase transitions on one-dimensional spatial random graphs, covering heavy-tailed degree distributions where earlier methods required exponential tails. The paper is generally well written, and the applications to long-range percolation (Can's conjecture for δ>2) and to sharp Gilbert-graph results are attractive. The proof idea of coupling the block process to a random walk in random environment is novel and potentially useful. However, the central RWRE coupling is not established as written, and the proof of one of the advertised applications contains a false inference.

major comments (4)
  1. [Section 3.1] The process Z, defined as the block index at jump times of Y, is not a Markov chain whose transition probabilities depend only on the block C_k. At the start of a sojourn in block k, the rightmost infected vertex is at the left boundary z_{k-1} if the previous block transition was from k-1 to k, but at the right boundary z_k-1 if it was from k+1 to k; the probabilities of the next exit being to k+1 versus k-1 differ in the two cases (for a two-vertex block with nearest-neighbour edges and λ<1, the right-exit probability from the left boundary is λ^2/((λ+1)^2-λ), while from the right boundary it is strictly larger). Hence the 'crucial observation' that Z is a random walk in a stationary ergodic environment is not established, and Ledrappier's Proposition 3.4 cannot be applied to the process as constructed. Moreover, since 0 is permanently infected in ξ†, the first block C_1 has no left exit, so the right-jump probability ω_1(λ) equals 1, contradicting the claim that ω_1(λ)↓0 as λ↓0. A regeneration construction (e.g., sampling the chain only at successive crossings of a fixed cut point, or enlarging the state to include the entry direction) is needed.
  2. [Proposition 3.4] The generator written there has ω_x as the coefficient of f(x-1), i.e., ω_x is the left-jump probability, but the recurrence criterion ∫ log((1-ω_1)/ω_1) dQ ≥ 0 is the criterion for ω_x being the right-jump probability. For the generator as stated, the correct criterion is ∫ log(ω_1/(1-ω_1)) dQ ≥ 0. Since the proof of Proposition 3.6 defines ω_k(λ) as the right-jump probability Pλ(Z_{k+1}=z+1|Z_k=z), the proposition as stated is inconsistent with its use and must be corrected.
  3. [Section 3.1, Eq. (11)] The paper claims that if the dominating process η returns to state {0} infinitely often, then the original process ξ^{{-1,0}} dies out. This implication is not immediate and is not proved: survival of ξ^{{-1,0}} is compatible with ξ†_t={0} infinitely often, since at those times the original process could itself be at {0}. A regenerative argument showing that each excursion of the original process away from {0} has a uniformly positive probability of dying out is required; as written, the deduction of extinction from recurrence of Z is incomplete.
  4. [Proof of Theorem 2.5] The step 'the probability that 0 is not covered by any ball of the Boolean model is positive if and only if ∫ rρ(dr)<∞; hence ... 0 has a positive probability of being a cut point' is not valid. A gap in the Boolean model around the origin does not prevent a long edge from crossing the link ℓ_0: two balls centered on opposite sides of the origin can overlap across it without either covering the origin. Moreover, for a deterministic lattice with constant radius 1, ∫ r dρ<∞ but every link has e(z)≥3, so 1-cut points do not exist. The application should verify the hypotheses of Theorem 2.2 directly, e.g., by showing P(e<∞)=1, rather than relying on the uncovered-origin event.
minor comments (4)
  1. [References] The reference list contains a duplicate entry: [27] and [28] are both Huang and Durrett, 'The contact process on random graphs and Galton Watson trees', ALEA 17 (2020).
  2. [Theorem 2.7] In Theorem 2.7, the displayed double integral in (7) is garbled: the lower limits of the two one-dimensional integrals should be 2^{-n-\mu n}; as typeset, the formula is not readable.
  3. [Section 3.3] The phrase 'spatially at most two dependent' is imprecise; the oriented edges are spatially dependent with bounded range and bounded degree, which is sufficient for the Liggett-Schonmann-Stacey theorem, but the number of dependent edges is not two in general.
  4. [Proposition 3.1] Proposition 3.1 refers to the sequence (C_k)_{k≥0}, but the blocks C_k are defined in Section 3.1 only for k≥1; please clarify the definition of C_0 or index the statement consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main result is derived from external criteria (Ledrappier, Kac, Harris) plus in-paper moment bounds; self-citations point to distinct prior theorems.

full rationale

The derivation of Theorem 2.2 is self-contained in the relevant sense. Assumptions (A.1)-(A.2) plus P(e<∞)=1 are used as hypotheses to obtain a positive density of K-cut points; this is not the conclusion λ_c>0. Proposition 3.1 applies Kac's lemma and the ergodic theorem to get E|C1|<∞ and the second-moment identity, and Proposition 3.3 derives E|E(C1)|<∞ from the edge-density identity (4), so the moment inputs to the key estimate (12) are proved inside the paper. The recurrence step is an application of Ledrappier's theorem [36,52], an external criterion, and the verification uses monotone convergence plus those internal moment bounds; no parameter is fitted and then renamed a prediction. The self-citations are to distinct prior results: [22] supplies cut-point existence for WDRCMs from condition (7), [29] supplies annulus-crossing estimates, and [41] supplies chemical-distance comparability; none contains the contact-process survival/extinction conclusion, so they count as independent support rather than circularity. A separate technical concern exists but is not circular: Section 3.1 asserts that 'Z is a random walk in a stationary ergodic random environment' without an explicit regeneration construction, and the jump probabilities of a coarse-grained Markov process need not be functions of the environment alone. This is a correctness gap, not an equivalence-by-construction, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central theorem rests on explicit assumptions (A.1), (A.2), and finiteness of e; all are stated and then verified for each application. No numerical parameters are fitted. The auxiliary objects (cut points, blocks, RWRE environment) are mathematical constructions, not new physical entities.

assumptions (7)
  • domain assumption Random graph G is stationary and ergodic under additive shifts (A.1).
    Assumed in Theorem 2.2; needed for a deterministic λ_c and for cut points to have positive density.
  • domain assumption The graph is sparse, Δ_G = E[deg(0)] < ∞ (A.2).
    Assumed in Theorem 2.2; used via the ergodic theorem for edge density and to prove finite expected edges per block.
  • domain assumption e, the number of edges crossing a typical link, is P-a.s. finite.
    This is the load-bearing cut-point condition; without it the block/RWRE decomposition fails.
  • standard math Ledrappier's recurrence criterion for nearest-neighbour random walks in stationary ergodic environments (Prop 3.4, from [36,52]).
    Used to convert block-process drift estimates into recurrence of the block process.
  • standard math Kac's lemma and cycle-stationarity of cut-point blocks (Prop 3.1).
    Gives finite expected block length and second moment; needed for the Ledrappier expectation bounds.
  • standard math Star-graph survival estimates from Huang and Durrett [28, Lemma 2.6].
    Used in the higher-dimensional no-threshold result, Theorem 2.8.
  • standard math Continuum percolation fact: a point is covered by at most finitely many balls iff ∫ r ρ(dr)<∞ [45].
    Basis for cut-point existence and local finiteness in augmented Gilbert graphs.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Phase transitions for contact processes on one-dimensional networks." pith.science (2026). https://pith.science/paper/NQ5DTK5S

@misc{pith2026250116858,
  author       = {Pith},
  title        = {Pith review of: Phase transitions for contact processes on one-dimensional networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQ5DTK5S}},
  note         = {Machine review of arXiv:2501.16858}
}
abstract

We study the survival/extinction phase transition for contact processes with quenched disorder. The disorder is given by a locally finite random graph with vertices indexed by the integers that is assumed to be invariant under index shifts and augments the nearest-neighbour lattice by additional long-range edges. We provide sufficient conditions that imply the existence of a subcritical phase and therefore the non-triviality of the phase transition. Our results apply to instances of scale-free random geometric graphs with any integrable degree distribution. The present work complements previously developed techniques to establish the existence of a subcritical phase on Poisson--Gilbert graphs and Poisson--Delaunay triangulations (M\'enard et al., Ann. Sci. \'Ec. Norm. Sup\'er., 2016), on Galton--Watson trees (Bhamidi et al., Ann. Probab., 2021) and on locally tree-like random graphs (Nam et al., Trans. Am. Math. Soc., 2022), all of which require exponential decay of the degree distribution. Two applications of our approach are particularly noteworthy: Firstly, for Gilbert graphs derived from stationary point processes on $\mathbb{R}$ marked with i.i.d. random radii, our results are sharp. We show that there is a non-trivial phase transition if and only if the graph is locally finite. Secondly, for independent Bernoulli long-range percolation on $\mathbb{Z}$, with coupling constants $J_{x,y}\asymp |x-y|^{-\delta}$, we verify a conjecture of Can (Electron. Commun. Probab., 2015) stating the non-triviality of the phase transition whenever $\delta>2$. We believe that the results are indicative of the behaviour of contact processes on spatial random graphs also in dimensions $d > 1$ as long as the degree distribution of the underlying network has at least finite $d$-th moment. We support this by proving that no phase transition exists if the $d$-th moment is infinite.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Concept-wise Attention for Fine-grained Concept Bottleneck Models

    cs.CV 2026-04 unverdicted novelty 7.0 of 10

    CoAt-CBM uses concept-wise visual queries and concept contrastive optimization to improve fine-grained image–concept alignment in CLIP-based concept bottleneck models.

  2. Phase transitions for contact processes on sparse random graphs via metastability and local limits

    math.PR 2025-05 conditional novelty 7.0 of 10

    For sparse locally convergent graph sequences, the exponential fast/slow threshold is bounded below by the local limit's survival threshold, and stretched-exponential extinction is possible below it.

Reference graph

Works this paper leans on

52 extracted references · 28 canonical work pages · cited by 2 Pith papers

  1. [22]

    Gracar, L

    P. Gracar, L. Lüchtrath, and C. Mönch.Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension. 2023. arXiv:2203.11966 [math.PR]

    arXiv 2023

  2. [1]

    Discontinuity of the percolation density in one-dimensional1/|x− y|2 percolation models

    M. Aizenman and C. M. Newman. “Discontinuity of the percolation density in one-dimensional1/|x− y|2 percolation models”.Comm. Math. Phys.107.4 (1986), pages 611–647

    work page 1986

  3. [2]

    Processes on unimodular random networks

    D. Aldous and R. Lyons. “Processes on unimodular random networks”.Electron. J. Probab.12 (2007). doi: 10.1214/EJP.v12- 463 . Extended and corrected version available athttps://rdlyons.pages.iu.edu/pdf/ urn.pdf

    work page doi:10.1214/ejp.v12- 2007

  4. [3]

    Ergodic theory on stationary random graphs

    I. Benjamini and N. Curien. “Ergodic theory on stationary random graphs”.Electron. J. Probab.17 (2012). doi: 10.1214/EJP.v17-2401

    doi:10.1214/ejp.v17-2401 2012

  5. [4]

    Survival and extinction of epidemics on random graphs with general degree

    S. Bhamidi, D. Nam, O. Nguyen, and A. Sly. “Survival and extinction of epidemics on random graphs with general degree”.Ann. Probab.49.1 (2021), pages 244–286.doi: 10.1214/20-AOP1451

    doi:10.1214/20-aop1451 2021

  6. [5]

    On the largest component of a hyperbolic model of complex networks

    M. Bode, N. Fountoulakis, and T. Müller. “On the largest component of a hyperbolic model of complex networks”. Electron. J. Combin.22.3 (2015). doi: 10.37236/4958

    work page doi:10.37236/4958 2015

  7. [6]

    Bollobás

    B. Bollobás. Random Graphs. Second Edition. Volume 73. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2001.doi: 10.1017/CBO9780511814068

    doi:10.1017/cbo9780511814068 2001

  8. [7]

    Spread-out percolation in Rd

    B. Bollobás, S. Janson, and O. Riordan. “Spread-out percolation in Rd”. Random Struct. Algorithms31.2 (2007), pages 239–246.doi: 10.1002/rsa.20175

    work page doi:10.1002/rsa.20175 2007

Show all 52 references
  1. [8]

    Contact process on one-dimensional long range percolation

    V. H. Can. “Contact process on one-dimensional long range percolation”. Electron. Commun. Probab.20 (2015). doi: 10.1214/ECP.v20-4461

    2015 doi

  2. [9]

    Contact processes on random graphs with power law degree distributions have critical value 0

    S. Chatterjee and R. Durrett. “Contact processes on random graphs with power law degree distributions have critical value 0”.Ann. Probab.37.6 (2009), pages 2332–2356.doi: 10.1214/09-AOP471. 20

    2009 doi

  3. [10]

    The average distance in a random graph with given expected degrees

    F. Chung and L. Lu. “The average distance in a random graph with given expected degrees”.Internet Math. 1.1 (2003), pages 91–113.doi: 10.1080/15427951.2004.10129081

    2003

  4. [11]

    Scale-free percolation

    M. Deijfen, R. van der Hofstad, and G. Hooghiemstra. “Scale-free percolation”.Ann. Inst. Henri Poincaré Probab. Stat.49.3 (2013), pages 817–838.doi: 10.1214/12-AIHP480

    2013 doi

  5. [12]

    A new proof of the sharpness of the phase transition for Bernoulli per- colation and the Ising model

    H. Duminil-Copin and V. Tassion. “A new proof of the sharpness of the phase transition for Bernoulli per- colation and the Ising model”.Comm. Math. Phys.343.2 (2016), pages 725–745.doi: 10.1007/s00220-015- 2480-z

    2016 doi

  6. [13]

    A new proof of the sharpness of the phase transition for Bernoulli perco- lation on Zd

    H. Duminil-Copin and V. Tassion. “A new proof of the sharpness of the phase transition for Bernoulli perco- lation on Zd”. Enseign. Math.62.1-2 (2016), pages 199–206.doi: 10.4171/LEM/62-1/2-12

    2016 doi

  7. [14]

    R. Durrett. Dynamics on Graphs. Book in Preparation, available underhttps://services.math.duke.edu/ ~rtd/DoG/allbook.pdf. 2024

    2024

  8. [15]

    L. R. Ford Jr. and D. R. Fulkerson.Flows in Networks. Princeton University Press, Princeton, NJ, 1962.doi: 10.1515/9780691273457

    1962 doi

  9. [16]

    P. A. Gomes, M. R. Hilário, B. N. B. de Lima, and T. Mountford.The extinction of the contact process in a one-dimensional random environment with long-range interactions. 2025. arXiv:2506.17444 [math.PR]

    2025

  10. [17]

    Subcritical regimes in the Poisson Boolean model of continuum percolation

    J.-B. Gouéré. “Subcritical regimes in the Poisson Boolean model of continuum percolation”. Ann. Probab. 36.4 (2008), pages 1209–1220.doi: 10.1214/07-AOP352

    2008 doi

  11. [18]

    The contact process on scale-free geometric random graphs

    P. Gracar and A. Grauer. “The contact process on scale-free geometric random graphs”.Stochastic Process. Appl. 173 (2024). doi: https://doi.org/10.1016/j.spa.2024.104360

    2024

  12. [19]

    The age-dependent random connection model

    P. Gracar, A. Grauer, L. Lüchtrath, and P. Mörters. “The age-dependent random connection model”.Queueing Syst. 93.3 (2019), pages 309–331.doi: 10.1007/s11134-019-09625-y

    2019 doi

  13. [20]

    Transience Versus Recurrence for Scale-Free Spatial Networks

    P. Gracar, M. Heydenreich, C. Mönch, and P. Mörters. “Transience Versus Recurrence for Scale-Free Spatial Networks”. In:Algorithms and Models for the Web Graph. Edited by B. Kamiński, P. Prałat, and P. Szufel. Lecture Notes in Computer Science. Cham: Springer International Pub...

    2020

  14. [21]

    Recurrence versus transience for weight-dependent random connection models

    P. Gracar, M. Heydenreich, C. Mönch, and P. Mörters. “Recurrence versus transience for weight-dependent random connection models”.Electron. J. Probab.27 (2022). doi: 10.1214/22-ejp748

    2022 doi

  15. [23]

    Percolation phase transition in weight-dependent random connection models

    P. Gracar, L. Lüchtrath, and P. Mörters. “Percolation phase transition in weight-dependent random connection models”. Adv. in Appl. Probab.53.4 (2021), pages 1090–1114.doi: 10.1017/apr.2021.13

    2021 doi

  16. [24]

    On continuum percolation

    P. Hall. “On continuum percolation”.Ann. Probab.13.4 (1985), pages 1250–1266

    1985

  17. [25]

    Contact interactions on a lattice

    T. E. Harris. “Contact interactions on a lattice”.Ann. Probab.2 (1974), pages 969–988.doi: 10.1214/aop/ 1176996493

    1974 doi

  18. [26]

    Distances in random graphs with finite variance degrees

    R. van der Hofstad, G. Hooghiemstra, and P. Van Mieghem. “Distances in random graphs with finite variance degrees”. Random Structures Algorithms27.1 (2005), pages 76–123.doi: 10.1002/rsa.20063

    2005 doi

  19. [28]

    The contact process on random graphs and Galton Watson trees

    X. Huang and R. Durrett. “The contact process on random graphs and Galton Watson trees”.ALEA Lat. Am. J. Probab. Math. Stat.17.1 (2020), pages 159–182.doi: 10.30757/alea.v17-07

    2020 doi

  20. [29]

    Jacob, B

    E. Jacob, B. Jahnel, and L. Lüchtrath. Subcritical annulus crossing in spatial random graphs. 2024. arXiv: 2411.10333 [math.PR]

    2024 arXiv

  21. [30]

    Robustness of scale-free spatial networks

    E. Jacob and P. Mörters. “Robustness of scale-free spatial networks”.Ann. Probab.45.3 (2017), pages 1680–

    2017

  22. [31]

    On the notion of recurrence in discrete stochastic processes

    M. Kac. “On the notion of recurrence in discrete stochastic processes”.Bull. Amer. Math. Soc.53 (1947), pages 1002–1010. doi: 10.1090/S0002-9904-1947-08927-8

    1947 doi

  23. [32]

    Variations on a theme by Mark Kac

    P. W. Kasteleyn. “Variations on a theme by Mark Kac”.J. Statist. Phys.46.5-6 (1987), pages 811–827.doi: 10.1007/BF01011143

    1987 doi

  24. [33]

    Shift-coupling of random rooted graphs and networks

    A. Khezeli. “Shift-coupling of random rooted graphs and networks”. In:Unimodularity in Randomly Generated Graphs. Volume 719. Contemp. Math. Amer. Math. Soc., Providence, RI, 2018, pages 175–211.doi: 10.1090/ conm/719/14474

    2018

  25. [34]

    Explosion in weighted hyperbolic random graphs and geometric inhomoge- neous random graphs

    J. Komjáthy and B. Lodewijks. “Explosion in weighted hyperbolic random graphs and geometric inhomoge- neous random graphs”.Stochastic Process. Appl.130.3 (2020), pages 1309–1367.doi: 10.1016/j.spa.2019. 04.014

    2020 doi

  26. [35]

    Last and M

    G. Last and M. Penrose. Lectures on the Poisson Process. Volume 7. Institute of Mathematical Statistics Textbooks. Cambridge University Press, Cambridge, 2018.doi: 10.1017/9781316104477. 21

    2018 doi

  27. [36]

    Quelques proprietes des exposants caracteristiques

    F. Ledrappier. “Quelques proprietes des exposants caracteristiques”. French. In:École d’Été de Probabilités de Saint-Flour XII - 1982. Edited by P. L. Hennequin. Berlin, Heidelberg: Springer Berlin Heidelberg, 1984, pages 305–396. doi: 10.1007/BFb0099434

    1982 doi

  28. [37]

    Domination by Product Measures

    T. M. Liggett, R. H. Schonmann, and A. M. Stacey. “Domination by Product Measures”.Ann. Probab.25.1 (1997), pages 71–95.doi: 10.1214/aop/1024404279

    1997

  29. [38]

    T. M. Liggett. Interacting Particle Systems. Volume 276. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1985. xv+488.doi: 10.1007/ 978-1-4613-8542-4

    1985

  30. [39]

    Stochastic Interacting Systems: Contact, Voter and Exclusion Processes.Volume324.Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    T.M.Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes.Volume324.Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-03990-8

    1999 doi

  31. [40]

    The contact process on random hyperbolic graphs: metastability and critical exponents

    A. Linker, D. Mitsche, B. Schapira, and D. Valesin. “The contact process on random hyperbolic graphs: metastability and critical exponents”.Ann. Probab.49.3 (2021), pages 1480–1514.doi: 10.1214/20-aop1489

    2021 doi

  32. [41]

    Lüchtrath

    L. Lüchtrath. All spatial random graphs with weak long-range effects have chemical distance comparable to Euclidean distance. 2024. arXiv:2412.12796 [math.PR]

    2024

  33. [42]

    Random walks and percolation on trees

    R. Lyons. “Random walks and percolation on trees”.Ann. Probab.18.3 (1990), pages 931–958.doi: 10.1214/ aop/1176990730

    1990

  34. [43]

    Random walks, capacity and percolation on trees

    R. Lyons. “Random walks, capacity and percolation on trees”.Ann. Probab.20.4 (1992), pages 2043–2088. doi: 10.1214/aop/1176989540

    1992

  35. [44]

    Lyons and Y

    R. Lyons and Y. Peres. Probability on Trees and Networks. Volume 42. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York, 2016.doi: 10.1017/9781316672815

    2016 doi

  36. [45]

    Meester and R

    R. Meester and R. Roy. Continuum Percolation. Volume 119. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996.doi: 10.1017/CBO9780511895357

    1996 doi

  37. [46]

    Percolation by cumulative merging and phase transition for the contact process on random graphs

    L. Ménard and A. Singh. “Percolation by cumulative merging and phase transition for the contact process on random graphs”.Ann. Sci. Éc. Norm. Supér. (4)49.5 (2016), pages 1189–1238.doi: 10.24033/asens.2307

    2016 doi

  38. [47]

    Critical value asymptotics for the contact process on random graphs

    D. Nam, O. Nguyen, and A. Sly. “Critical value asymptotics for the contact process on random graphs”. Trans. Amer. Math. Soc.375.6 (2022), pages 3899–3967.doi: 10.1090/tran/8399

    2022 doi

  39. [48]

    The branching random walk and contact process on Galton-Watson and nonhomogeneous trees

    R. Pemantle and A. M. Stacey. “The branching random walk and contact process on Galton-Watson and nonhomogeneous trees”.Ann. Probab.29.4 (2001), pages 1563–1590.doi: 10.1214/aop/1015345762

    2001

  40. [49]

    On a continuum percolation model

    M. D. Penrose. “On a continuum percolation model”.Adv. in Appl. Probab.23.3 (1991), pages 536–556.doi: 10.2307/1427621

    1991 doi

  41. [50]

    Long range percolation in one dimension

    L. S. Schulman. “Long range percolation in one dimension”.J. Phys. A16.17 (1983), pages L639–L641.doi: 10.1088/0305-4470/16/17/001

    1983 doi

  42. [51]

    On time- and cycle-stationarity

    H. Thorisson. “On time- and cycle-stationarity”. Stochastic Process. Appl.55.2 (1995), pages 183–209.doi: 10.1016/0304-4149(94)00038-U

    1995 doi

  43. [52]

    Random walks in random environment

    O. Zeitouni. “Random walks in random environment”. In: Lectures on Probability Theory and Statistics. Volume 1837. Lecture Notes in Math. Springer, Berlin, 2004, pages 189–312.doi: 10.1007/978-3-540-39874- 5\_2. 22

    2004 doi

  44. [1722]

    doi: 10.1214/16-AOP1098

    doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.