Pith. sign in

REVIEW 3 major objections 3 minor 1 cited by

Nonamenable Poisson zoo

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read This paper proves that on nonamenable unimodular transitive graphs, random-walk worms with infinite squared length percolate at every positive intensity, and that the same holds for arbitrary lattice animals on nonamenable free products.

desk verdict Strong new results for free products and a nice uniqueness example, but the worm theorem on general nonamenable graphs has a false inequality at (3.27) that currently sinks Theorem 1.1. read the letter →

arxiv 2505.07145 v1 pith:SCNMW5XA submitted 2025-05-11 math.PR math-phmath.GRmath.MP

Gábor Pete , Sándor Rokob This is my paper · ORCID
classification math.PRmath-phmath.GRmath.MP MSC 60K3582B4137A20
keywords Poissonzoolatticeanimalspercolationnonamenablegraphsrandomwalkwormsfreeproductsphasetransitionunimodularity
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

This paper tries to establish that the Poisson zoo can percolate at arbitrarily small intensity on nonamenable transitive graphs, provided the random lattice animals have infinite expected squared size but finite expected size. For random-walk worms this is proved on every nonamenable unimodular transitive graph; for arbitrary animals it is proved on nonamenable unimodular free products. The interest is that these models have infinite clusters despite arbitrarily low density, a behaviour impossible for Bernoulli percolation, and they give sparse factor-of-i.i.d. percolations relevant to measurable cost. A companion example on $T_d\times\mathbb{Z}^5$ has a unique infinite cluster at every positive intensity, with finite first moment. The fully general case -- arbitrary animals on arbitrary nonamenable Cayley graphs -- is left open.

What carries the argument

The carrying mechanism is an exploration process that 'fattens' the already explored cluster through its exterior boundary, adding only animals that touch the boundary and avoid the old set. For worms, the expected size of the fat is controlled by random-walk capacity, which is comparable to volume on nonamenable graphs; unimodularity produces size-biased first moments, a corollary neglects multiplicities, and a first-moment lower bound combined with a second-moment upper bound feeds a concentration lemma proving exponential growth forever. For free products, the exploration embeds a Galton-Watson branching process using the fact that every vertex is a cutpoint, with a sprinkling step that repairs the parity issue when growth is available in only one factor of the product.

What would settle it

Compute both sides of inequality (3.27) for a worm process on a regular tree, taking $A$ to be a short path and $B$ to be two adjacent exterior vertices, with a worm length distribution that lets one worm touch both vertices. If the left side is strictly smaller than the right side for some $R$, the stated lower bound lacks its proof; if a corrected argument restores the inequality, the conclusion of the worm theorem follows. A direct counterexample to the theorem would be a nonamenable unimodular transitive graph and worm lengths with $\mathbb{E}L^2=\infty$ but $\lambda_c>0$.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for any nonamenable unimodular transitive graph $G$, the worms model -- animals that are traces of simple random walks of random finite length $L$ -- has $\lambda_c=0$ whenever $\mathbb{E}L<\infty$ and $\mathbb{E}L^2=\infty$. Theorem 1.2 extends the same conclusion to arbitrary lattice animal measures with $\mathbb{E}|H|^2=\infty$ when $G$ is a nonamenable unimodular free product. Proposition 1.4 constructs a zoo on $\mathbb{T}_d\times\mathbb{Z}^5$ with $\mathbb{E}|H|<\infty$ and a unique infinite cluster for every $\lambda>0$. The paper leaves open whether $\lambda_c=0$ holds for every animal measure with infinite second moment on every nonamenable Cayley graph.

Load-bearing premise

The worm proof assumes that worms touching several boundary vertices can be counted vertex-by-vertex without loss, so that the total fat is at least the sum of single-vertex fats; the displayed inequality does this, but the definitions do not discard multi-touching worms.

Editorial extensions

If this is right

  • If Theorem 1.1 is correct, every nonamenable unimodular transitive graph admits worm zoos with $\lambda_c=0$ for every length distribution with finite mean and infinite second moment.
  • If Theorem 1.2 is correct, on nonamenable free products the critical intensity is zero for every animal measure with infinite second moment, independent of animal shape.
  • Arbitrarily sparse Poisson zoos can percolate on nonamenable graphs, so in this model low density does not prevent infinite clusters.
  • The $\mathbb{T}_d\times\mathbb{Z}^5$ example shows $\lambda_u=0$ with finite first moment: uniqueness of the infinite cluster can also be immediate.
  • The general Question 1.3 -- arbitrary animals on arbitrary nonamenable Cayley graphs -- remains open.

Reading between the lines

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

  • A testable extension suggested by the method is that worms with infinite second moment may also have $\lambda_c=0$ on other transient graphs where capacity is not linear, such as high-dimensional amenable lattices; the linear-capacity proof would not transfer directly.
  • The concentration lemma that turns bounded variance into survival is a transferable tool: any Poisson soup whose growth increments satisfy comparable first and second moment bounds should exhibit permanent supercriticality.
  • The uniqueness example points toward constructing sparse factor-of-i.i.d. percolations with a unique infinite cluster on groups with vanishing first $\ell^2$-Betti number, which by known criteria would imply cost $1$; the paper raises this as Question 1.5 but does not resolve it.
  • If the disputed fattening inequality is repaired, the worm theorem would be proven; if not, the theorem may still be true but would need a different exploration bound.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies the Poisson zoo model on transitive graphs, in which i.i.d. Poisson(λ) copies of random finite connected subsets (lattice animals) are placed with intensity λ. The three main claims are: (1) for any nonamenable unimodular transitive graph, worms of random length with infinite second moment have λ_c=0 (Theorem 1.1, proved as Theorem 2.17); (2) for any nonamenable free product of transitive graphs and any animal measure with infinite second moment, λ_c=0 (Theorem 1.2); and (3) on T_d×Z^5 there is a zoo with finite first moment and a unique infinite cluster for every λ>0 (Proposition 1.4). The proofs are built on exploration processes: a branching process on free products, and a fattening exploration process on general nonamenable graphs, supported by first- and second-moment bounds on the newly occupied set. I find a load-bearing false inequality in the worm proof, Corollary 3.9, inequality (3.27), and an additional invalid algebraic step in the same corollary. As written, the proof of Theorem 1.1 is not established.

Significance. If correct, the results would be a substantial advance: they resolve the worm case for all nonamenable unimodular transitive graphs, give the first general-animal result on free products, and provide an appealing example of immediate uniqueness with a sparse FIID cluster. The paper uses elegant tools, including size-biasing via unimodular mass transport, capacity bounds on nonamenable graphs, and Poisson point process restrictions; these parts are mostly well executed. However, the worm theorem is one of the two headline results, and its proof currently rests on a false inequality and an incorrect algebraic passage. The free-product theorem and the uniqueness example are independent of the worm argument, but the manuscript as a whole cannot be accepted until the worm proof is repaired.

major comments (3)
  1. [§3.3, Corollary 3.9, Eq. (3.27)] Inequality (3.27) is false as written. By Definition 3.1, the left-hand side counts each animal of volume at most R that hits B and avoids A\B exactly once, while each summand on the right counts the same animal once for every vertex of B it hits. For an animal whose trace intersects B in k vertices, the contribution to the right-hand side is k times its contribution to the left-hand side, so the correct inequality is the reverse, not (3.27). The sentence in the proof saying that worms hitting several vertices of B are "thrown away" does not correspond to any restriction present in (3.1). This is load-bearing: Corollary 3.9 is the only first-moment lower bound used to verify hypothesis (4.27) of Proposition 4.8, and without (3.27) the sum over B of the single-vertex bounds from Lemma 3.8 does not yield a lower bound on the total occupation measure of the unrestricted process. Thus the proof of Theorem 2.17, and hence of Theorem 1.1, collapses at this point unless a correct argument replacing (3.27) is supplied.
  2. [§3.3, Corollary 3.9, Eq. (3.29)] The displayed chain in (3.29) is also algebraically invalid. From (2.20) one gets Σ p_x(p_x−ε) ≥ (1−ρ)^2|A| − (|∂extA|−|B|) − ε|A|. The next line claims this is at least ((1−ρ)^2−ε)(1+h)|A| − (|∂extA|−|B|), but the difference between the claimed lower bound and the obtained bound is h[(1−ρ)^2−ε]|A|, which is positive for h>0 and ε<(1−ρ)^2. Therefore the factor (1+h) in (3.29), and consequently the factor (1+h)/2 in (3.26), is not derived from Corollary 2.9. For graphs with h>1, this factor asks for a strictly stronger capacity lower bound than Lemma 2.8 provides. The constants in (3.26) and in hypothesis (4.27) therefore need to be reworked independently of the issue in (3.27).
  3. [§5, Proposition 1.4] The proof of Proposition 1.4 relies on the in-preparation reference [23] for indistinguishability of infinite clusters in Poisson zoos. Since no proof of this statement appears in the manuscript, the proposition is currently conditional on an external unpublished result. The authors should either include a proof of the needed indistinguishability statement, give a publicly available reference, or explicitly mark Proposition 1.4 as conditional.
minor comments (3)
  1. [§3.3, Lemma 3.8, Eqs. (3.19) and (3.21)] The condition H∩A={x} in (3.19) and (3.21) should be H∩A=∅, since x∈∂extA and hence x∉A. As written, the event is empty, although the subsequent estimates clearly use the intended avoidance condition.
  2. [§2.5, Theorem 2.17] The theorem states only E[L^2]=∞, while the abstract and Lemma 2.16 also impose E[L]<∞. The missing condition should be stated explicitly, even though the case E[L]=∞ is trivial by Lemma 2.14.
  3. [§3.3, proof of Lemma 3.8] There is a typo "substitue" in the sentence before (3.25); it should be "substitute."

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the main derivations are self-contained, with only minor non-load-bearing self-citations.

full rationale

The derivation chain for Theorem 1.1/2.17 runs from the moment identity Lemma 2.16, the size-biasing Lemma 3.2, the worm first-moment bound Lemma 3.8, Corollary 3.9, the variance bound Lemma 3.10, and Proposition 4.8; the hypothesis E[L^2]=∞ enters only as lim_{R→∞} E[L^2 1_{L≤R}] = ∞, and the conclusion λ_c=0 is not used anywhere as an input. Theorem 1.2 is similarly driven by Corollary 3.7 and Claims 4.3–4.4, with m_2^R→∞ supplying supercriticality; no fitted parameter is later renamed a prediction. Proposition 1.4 uses the published external result [46] to supply the input worm measure on Z^5 and the external indistinguishability theorem [39] for the insertion-tolerant zoo; the in-preparation self-citation [23] ('with a simpler proof in [23] that also includes any Poisson zoo') is not load-bearing here because [39] already covers the insertion-tolerant case. The questionable inequality in Corollary 3.9 at (3.27) is a possible correctness gap, not a circularity: even if a worm hitting several B-vertices is counted once on the left and several times on the right, that error would undermine a lower bound rather than make the theorem equivalent to its assumptions. Accordingly, no constructed derivation reduces to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard probabilistic machinery and on two prior external results: the Z^5 worm result of [46] and the indistinguishability theorem of [23], the latter still in preparation. No fitted constants or ad hoc parameters appear, and no new physical or mathematical entities are introduced.

assumptions (5)
  • domain assumption Mass transport principle / unimodularity of the group action
    Used in Lemma 2.14 and Lemma 3.2 to make expectations of occupation measures equal λ times first and second truncated moments; every Cayley graph satisfies it, but non-unimodular transitive graphs fail, as shown in Remark 3.3.
  • standard math Spectral radius ρ(G)<1 characterizes nonamenability
    Used in (2.10) and Lemma 2.8 to make capacity of finite sets linear in volume, which is the engine of the worm exploration.
  • standard math Linear speed of random walk on nonamenable transitive graphs (Theorem 2.6 and Corollary 2.7)
    Used in Lemma 2.16 and Lemma 3.8 to compare worm trace size to walk length with positive probability.
  • domain assumption For worms on Z^5 with finite first moment and λ_c=0, provided by Rath and Rokob [46]
    Input to Proposition 1.4's construction of a zoo with a unique infinite cluster.
  • domain assumption Indistinguishability of infinite clusters for insertion-tolerant invariant percolation and Poisson zoo, cited to [23] as in preparation
    Used at the end of Proposition 1.4 to rule out extra infinite clusters; the cited work by El Alami, Pete, and Timár is not publicly available.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Nonamenable Poisson zoo." pith.science (2026). https://pith.science/paper/SCNMW5XA

@misc{pith2026250507145,
  author       = {Pith},
  title        = {Pith review of: Nonamenable Poisson zoo},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SCNMW5XA}},
  note         = {Machine review of arXiv:2505.07145}
}
abstract

In the Poisson zoo on an infinite Cayley graph $G$, we take a probability measure $\nu$ on rooted finite connected subsets, called lattice animals, and place i.i.d. Poisson($\lambda$) copies of them at each vertex. If the expected volume of the animals w.r.t. $\nu$ is infinite, then the whole $G$ is covered for any $\lambda>0$. If the second moment of the volume is finite, then it is easy to see that for small enough $\lambda$ the union of the animals has only finite clusters, while for $\lambda$ large enough there are also infinite clusters. Here we show that: 1. If $G$ is a nonamenable free product, then for ANY $\nu$ with infinite second but finite first moment and any $\lambda>0$, there will be infinite clusters, despite having arbitrarily low density. 2. The same result holds for ANY nonamenable $G$, when the lattice animals are worms: random walk pieces of random finite length. It remains open if the result holds for ANY nonamenable Cayley graph with ANY lattice animal measure $\nu$ with infinite second moment. 3. We also give a Poisson zoo example $\nu$ on $\mathbb{T}_d \times \mathbb{Z}^5$ with finite first moment and a UNIQUE infinite cluster for any $\lambda>0$.

Figures

Figures reproduced from arXiv: 2505.07145 by the authors.

Figure 1.1
Figure 1.1. Exploring a cluster: at any given stage En−1, there are many exposed vertices on its boundary where new worms (the green trajectories) can touch it, creating a much larger En. The key advantage of nonamenable transitive graphs compared to Z d , high d, is that the random walk capacity of any finite set S ⊂ V (G) is linear in the volume |S|; see Lemma 2.8. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Two examples of free products of graphs. [PITH_FULL_IMAGE:figures/full_fig_p004_1_2.png] view at source ↗
Figure 2.1
Figure 2.1. A parity issue. On the free product Z ⋆ Z, generated by the letters a and b, take a ν with m2(ν) = ∞ that is supported only on words in one of the generators, say a (the “horizontal” red generator in the picture). It may happen that we have already explored a large connected set, shaded green in the picture, which has of course a large total exterior boundary, but only a small one in the b direction. But only these … view at source ↗
Figures from the paper (1 more)
Figure 4.1
Figure 4.1. Figure 4.1: The fat Cn+1 is the growth through Bn, yielding En+1, and then Bn+1 is the new part of the exterior boundary of En+1. An important consequence of this choice of Bn can be described once we define the filtration corresponding to the exploration process: Fn := σ ({Ek} …

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. (Non-)coincidence of critical parameters for Poisson Zoos and Loop Soup Percolation on $\mathbb{Z^d}$, $d > 4$, and $\mathbb{T}_d$,$ d \ge 3$

    math.PR 2026-08 accept novelty 7.0 of 10

    Loop soup percolation on Z^d, d≥5, has discrete threshold strictly above the cable system threshold 1/2; on trees, Poisson zoo percolation and susceptibility thresholds coincide, and the model is infinitesimally sensi...

Reference graph

Works this paper leans on

58 extracted references · 53 canonical work pages · cited by 1 Pith paper

  1. [23]

    El Alami, G

    D. El Alami, G. Pete, and ´A. Tim´ ar (2025)In preparation

    work page 2025

  2. [1]

    Ab´ ert and B

    M. Ab´ ert and B. Weiss (2013) Bernoulli actions are weakly contained in any free action. Ergodic Theory and Dynamical Systems 33, 323–333

    work page 2013

  3. [2]

    Babson and I

    E. Babson and I. Benjamini (1999) Cut sets and normed cohomology, with applications to percolation. Proc. Amer. Math. Soc. 127, 589–597

    work page 1999

  4. [3]

    Bandyopadhyay, J

    A. Bandyopadhyay, J. Steif and ´A. Tim´ ar (2010) On the cluster size distribution for per- colation on some general graphs. Revista Matem´ atica Iberoamericana26, 529–550

    work page 2010

  5. [4]

    Benjamini, J

    I. Benjamini, J. Jonasson, O. Schramm and J. Tykesson (2009) Visibility to infinity in the hyperbolic plane, despite obstacles. ALEA Latin Amer. J. Prob. Math. Statist. 6, 323–342

    work page 2009

  6. [5]

    Benjamini, R

    I. Benjamini, R. Lyons, Y. Peres and O. Schramm (1999) Group-invariant percolation on graphs. Geom. Funct. Anal. 9, 29–66

    work page 1999

  7. [6]

    Benjamini and O

    I. Benjamini and O. Schramm (1996) Percolation beyond Zd, many questions and a few answers. Electronic Communications in Probability 1, 71–82

    work page 1996

  8. [7]

    Benjamini, A

    I. Benjamini, A. Nachmias, and Y. Peres (2011) Is the critical percolation probability local? Probab. Theory Relat. Fields 149, 261—269

    work page 2011

Show all 58 references
  1. [8]

    Borb´ enyi, B

    M. Borb´ enyi, B. R´ ath, and S. Rokob (2023) Random interlacement is a factor of iid. Electronic Journal of Probability 28, 1–45

    2023

  2. [9]

    Bowen (2019) Finitary random interlacements and the Gaboriau–Lyons problem

    L. Bowen (2019) Finitary random interlacements and the Gaboriau–Lyons problem. Geom. Funct. Anal. 29, 659-–689

    2019

  3. [10]

    E. I. Broman and J. Tykesson (2016) Connectedness of Poisson cylinders in Euclidean space. Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques52, 102–126

    2016

  4. [11]

    E. I. Broman and J. Tykesson (2015) Poisson cylinders in hyperbolic space. Electronic Journal of Probability 20, 1–25

    2015

  5. [12]

    Z Cai, E. B. Procaccia, and Y. Zhang (2022) Continuity and uniqueness of percolation critical parameters in finitary random interlacements. Elect. Journ. Probab. 27, 1–46

    2022

  6. [13]

    Chang (2021) Bernoulli hyper-edge percolation on Zd

    Y. Chang (2021) Bernoulli hyper-edge percolation on Zd. arXiv:2101.06082

    2021 arXiv

  7. [14]

    Chang and A

    Y. Chang and A. Sapozhnikov (2016) Phase transition in loop percolation. Probab. Theory Relat. Fields 164, 979–1025

    2016

  8. [15]

    C. F. Coletti, D. Mirana and S. P. Grynberg (2020) Boolean percolation on doubling graphs. Journal of Statistical Physics 178, 814–831

    2020

  9. [16]

    Dembin and V

    B. Dembin and V. Tassion (2022) Almost sharp sharpness for Poisson Boolean percolation. arXiv:2209.00999

    2022 arXiv

  10. [17]

    Dewan and S

    V. Dewan and S. Muirhead (2023) Mean-field bounds for Poisson-Boolean percolation. Electron. J. Probab. 28, 1–24

    2023

  11. [18]

    Drewitz, A

    A. Drewitz, A. Pr´ evost, and P.-F. Rodriguez (2024) Geometry of Gaussian free field sign clusters and random interlacements. Probability Theory and Related Fields , 96 pages

    2024

  12. [19]

    Drewitz, B

    A. Drewitz, B. R´ ath and A. Sapozhnikov (2014)An introduction to Random Interlacements. SpringerBriefs in Mathematics, Springer. 38

    2014

  13. [20]

    Duminil-Copin, S

    H. Duminil-Copin, S. Goswami, A. Raoufi, F. Severo, and A. Yadin (2020) Existence of phase transition for percolation using the Gaussian free field. Duke Mathematical Journal 169, 3539–3563

    2020

  14. [21]

    Dumini-Copin, S

    H. Dumini-Copin, S. Goswami, P.-F. Rodriguez and F. Severo (2023) Equality of critical parameters for percolation of Gaussian free field level sets. Duke Math. J. 172, 839–913

    2023

  15. [22]

    P. Easo, F. Severo, and V. Tassion (2024) Counting minimal cutsets and pc < 1. arXiv:2412.04539

    2024

  16. [24]

    M. P. Forsstr¨ om, N. Gantert and J. E. Steif (2024) Poisson representable processes. arxiv:2401.13412v1

    2024 arXiv

  17. [25]

    Fr¸ aczyk, S

    M. Fr¸ aczyk, S. Mellick, A. Wilkens (2023) Poisson-Voronoi tessellations and fixed price in higher rank. arXiv:2307.01194

    2023 arXiv

  18. [26]

    Gaboriau (2000) Coˆ ut des relations d’´ equivalence et des groupes.Inv

    D. Gaboriau (2000) Coˆ ut des relations d’´ equivalence et des groupes.Inv. math. 139, 41–98

    2000

  19. [27]

    Gaboriau (2010) Orbit equivalence and measured group theory

    D. Gaboriau (2010) Orbit equivalence and measured group theory. Proc. ICM 2010 Hyder- abad, World Scientific

    2010

  20. [28]

    Gaboriau and R

    D. Gaboriau and R. Lyons (2009) A measurable-group-theoretic solution to von Neumann’s problem. Inventiones mathematicae 177, 533–540

    2009

  21. [29]

    Gou´ er´ e (2008) Subcritical regimes in the Poisson Boolean model of continuum per- colation

    J.–B. Gou´ er´ e (2008) Subcritical regimes in the Poisson Boolean model of continuum per- colation. The Annals of Probability 36, 1209–1220

    2008

  22. [30]

    Gracar, M

    P. Gracar, M. Korfhage, and P. M¨ orters (2024) Robustness in the Poisson Boolean model with convex grains arXiv:2410.13366

    2024 arXiv

  23. [31]

    Greb´ ık and K

    J. Greb´ ık and K. Recke (2025) Poisson-Voronoi percolation in higher rank. arXiv:2504.02435

    2025 arXiv

  24. [32]

    H¨ aggstr¨ om (1997) Infinite clusters in dependent automorphism invariant percolation on trees

    O. H¨ aggstr¨ om (1997) Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25, 1423–1436

    1997

  25. [33]

    Hutchcroft and G

    T. Hutchcroft and G. Pete (2020) Kazhdan groups have cost 1. Inv. Math. 221, 873–891

    2020

  26. [34]

    J. F. C. Kingman (1993) Poisson processes. Oxford Studies in Probability, vol. 3. Oxford University Press, New York

    1993

  27. [35]

    Levitt (1995) On the cost of generating an equivalence relation

    G. Levitt (1995) On the cost of generating an equivalence relation. Ergodic Theory and Dynamical Systems 15, 1173–1181

    1995

  28. [36]

    Lupu (2016) From loop clusters and random interlacements to the free field.Ann

    T. Lupu (2016) From loop clusters and random interlacements to the free field.Ann. Probab. 44, 2117–2146

    2016

  29. [37]

    Lyons (2017) Factors of IID on trees

    R. Lyons (2017) Factors of IID on trees. Combin. Probab. Comput. 26, 285–300

    2017

  30. [38]

    Lyons and Y

    R. Lyons and Y. Peres (2016) Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics, 42. Cambridge University Press, New York

    2016

  31. [39]

    Lyons and O

    R. Lyons and O. Schramm (1999) Indistinguishability of percolation clusters. Ann. Probab. 27, 1809–1836

    1999

  32. [40]

    Meester and R

    R. Meester and R. Roy (1996) Continuum Percolation, Cambridge University Press. 39

    1996

  33. [41]

    Mu and A

    Y. Mu and A. Sapozhnikov (2023) Uniqueness of the infinite connected component for the vacant set of random interlacements on amenable transient graphs. Electronic Communi- cations in Probability 28, 1–9

    2023

  34. [42]

    Pak and T

    I. Pak and T. Smirnova-Nagnibeda (2000) On non-uniqueness of percolation on nona- menable Cayley graphs Comptes Rendus de l’Acad. Sci. Series I Math. 330, 495–500

    2000

  35. [43]

    Pete Probability and Geometry on Groups: Lecture notes for a graduate course

    G. Pete Probability and Geometry on Groups: Lecture notes for a graduate course . Book in preparation, available at https://math.bme.hu/ gabor/PGG.pdf

  36. [44]

    Pete, ´A

    G. Pete, ´A. Tim´ ar, S.¨O. Stef´ ansson, I. Bonamassa, and M. P´ osfai (2024) Physical networks as network-of-networks. Nature Communications 15, article number 4882

    2024

  37. [45]

    E. B. Procaccia, J. Ye, and Y. Zhang (2021) Percolation for the finitary random interlace- ments. ALEA, Lat. Am. J. Probab. Math. Stat. 18, 265–287

    2021

  38. [46]

    R´ ath and S

    B. R´ ath and S. Rokob (2022) Percolation of worms. Stochastic Processes and their Appli- cations 152, 233–288

    2022

  39. [47]

    S. I. Resnick (2008) Extreme values, regular variation and point processes . Springer Series on Operations Research and Financial Engineering. Springer, New York

    2008

  40. [48]

    Teixeira and D

    A. Teixeira and D. Ungaretti (2017) Ellipses percolation. Journ. Stat. Phys. 38, 369–393

    2017

  41. [49]

    Teixeira (2009) Interlacement percolation on transient weighted graphs.Electronic Jour- nal of Probability 14, 1604–1627

    A. Teixeira (2009) Interlacement percolation on transient weighted graphs.Electronic Jour- nal of Probability 14, 1604–1627

    2009

  42. [50]

    Teixeira and J

    A. Teixeira and J. Tykesson (2013) Random interlacements and amenability. The Annals of Applied Probability 23, 923–956

    2013

  43. [51]

    Thom (2015) A remark about the spectral radius

    A. Thom (2015) A remark about the spectral radius. Int. Math. Res. Not. 2015, 2856–2864

    2015

  44. [52]

    ´A. Tim´ ar. (2007) Cutsets in infinite graphs.Combin. Probab. & Comput. 16, 159–166

    2007

  45. [53]

    Tykesson and D

    J. Tykesson and D. Windisch (2012) Percolation in the vacant set of Poisson cylinders. Probability Theory and Related Fields 154, 165–191

    2012

  46. [54]

    Tykesson and P

    J. Tykesson and P. Calka (2013) Asymptotics of visibility in the hyperbolic plane.Advances in Applied Probability 45, 332–350

    2013

  47. [55]

    Tykesson (2007) The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space

    J. Tykesson (2007) The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space. Elect. Journ. Probab. 12, 1379–1401

    2007

  48. [56]

    Tykesson (2009) Continuum Percolation at and above the Uniqueness Threshold on Homogeneous Spaces

    J. Tykesson (2009) Continuum Percolation at and above the Uniqueness Threshold on Homogeneous Spaces. Journal of Theoretical Probability 22, 402–417

    2009

  49. [57]

    Sznitman (2010) Vacant set of random interlacements and percolation

    A.-S. Sznitman (2010) Vacant set of random interlacements and percolation. Annals of Mathematics 171, 2039–2087

    2010

  50. [58]

    Woess (2000) Random Walks on Infinite Graphs and Groups

    W. Woess (2000) Random Walks on Infinite Graphs and Groups . Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge. 40

    2000

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.