Percolation phase transitions for the SIR model with random powers

Regine L\"offler

REVIEW 3 major objections 5 minor 19 references

Percolation phase transitions for the SIR model with random powers

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A SINR network whose users transmit with random powers percolates at large user intensities whenever the power can exceed a large threshold with positive probability and has finite exponential moments.

desk verdict The new supercritical theorem in Chapter 4 is plausible but the written proof has a real gap in Step 3; the subcritical half is clean and the thesis is worth engaging, but a referee should insist on fixing the transfer. read the letter →

arxiv 1908.07375 v1 pith:BKZHW4EO submitted 2019-08-12 math.PR

Regine Löffler This is my paper · ORCID

classification math.PR MSC 60K3560D0560G55

keywords continuumpercolationSINRmodelrandomtransmissionpowersPoissonpointprocessphasetransitioninterferenceshotnoiseGilbertgraph

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies a telecommunication network in which users are placed by a Poisson point process in $\mathbb{R}^d$, $d \ge 2$, and each user transmits with an independent random power. It asks when an infinite multi-hop communication component exists in the graph where two users are linked if their signal-to-interference-and-noise ratio exceeds a threshold in both directions. It proves a subcritical phase when the path-loss function decays like $r^{-\beta}$ and the power tail decays like $r^{-\alpha}$ with $\alpha\beta > 2d-1$: for small intensity there is no infinite component. It proves a supercritical phase when the power exceeds $N_0\tau/\ell(0)$ with positive probability and some exponential moment is finite: for all sufficiently large intensities there is a positive interference tolerance $\gamma^*(\lambda)$ such that percolation holds for every $\gamma \le \gamma^*(\lambda)$. The result removes the bounded-power restriction of earlier work and extends the model from $\mathbb{R}^2$ to all dimensions $d \ge 2$.

What carries the argument

The proof constructs a site percolation on $\mathbb{Z}^d$. A site $z$ is good if some Poisson point in the unit cube near $z$ has random power above $r$, and all Poisson points in a larger cube are connected inside an even larger cube by the Gilbert graph with radius $\delta = \ell^{-1}(N_0\tau/r)/2$; a site is open if it is good and its weighted shot noise $\tilde{I}_6(z) = \sum_i \rho_i \ell_6(|z-X_i|)$ is at most $M$. The bridge from lattice to continuum is the SINR lower bound: two points at distance at most $\delta$ with powers above $r$ and total interference at most $M$ satisfy $\rho\ell(\delta)/(N_0+\gamma M) > \tau$ whenever $\gamma \le \gamma^* = (N_0/M)(\ell(\delta)/\ell(2\delta)-1)$, since $\ell(2\delta) = N_0\tau/r$. Thus percolation of open lattice sites forces an infinite connected component in the SINR graph.

What would settle it

Test the boundary value $r = N_0\tau/\ell(0)$. With $\delta = 0$, the Gilbert graph $g_0$ has no edges between distinct points, so any unit cube containing two Poisson points is not good; for large $\lambda$ such configurations occur with probability close to 1, making $\mathbb{P}(A(o)=0) \to 1$, the opposite of Proposition 4.8's conclusion. A simulation or exact computation of $\mathbb{P}(A(o)=0)$ as $\lambda \to \infty$ at this boundary would settle whether the theorem's non-strict statement can be saved.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.5. In the SINR model with random powers on a Poisson point process of intensity $\lambda$ in $\mathbb{R}^d$, $d \ge 2$, assume the nonnegative power variable $\rho$ is not a.s. zero, has finite expectation, satisfies $\mathbb{P}(\rho > r) > 0$ for some $r \ge N_0\tau/\ell(0)$, and has finite exponential moment $\mathbb{E}[e^{\alpha\rho}] < \infty$ for some $\alpha > 0$. Then for every sufficiently large $\lambda$ there exists $\gamma^*(\lambda) > 0$ such that the SINR graph percolates for every $\gamma \le \gamma^*(\lambda)$. Together with Corollary 4.4, which gives a subcritical phase when $\ell(r) \le r^{-\beta}$ and $\mathbb{P}(\rho > r) \le r^{-\alpha}$ with $\alpha\beta > 2d-1$, this yields a phase transition under conditions that allow unbounded, heavy-tailed powers.

Load-bearing premise

The supercritical-phase proof needs a power threshold strictly above $N_0\tau/\ell(0)$ so that the auxiliary radius $\delta = \ell^{-1}(N_0\tau/r)/2$ is positive; the theorem states only $r \ge N_0\tau/\ell(0)$, and the boundary case is where the good-site construction loses its footing.

Editorial extensions

If this is right

An unbounded random power distribution can be used in the supercritical regime, provided its tail has enough positive mass at large powers and its exponential moment is finite; this removes the earlier boundedness condition on $\rho$.
For large user densities, the network tolerates a strictly positive interference level $\gamma^*(\lambda)$ while still containing an infinite communication component, so the phase transition in $\gamma$ occurs at a positive value.
The quantitative subcritical criterion $\alpha\beta > 2d-1$ links the decay of the path-loss function and the decay of the power tail; both can be heavy as long as their product decays fast enough.
The statements hold for every dimension $d \ge 2$, so the $d=2$ restriction of the earlier bounded-power model is removed.

Reading between the lines

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

A likely repair for the boundary case is to replace $r \ge N_0\tau/\ell(0)$ by $r > N_0\tau/\ell(0)$: whenever $\mathbb{P}(\rho > r) > 0$ holds at the boundary, continuity from below yields a slightly larger threshold with positive mass, so the auxiliary radius $\delta$ can be made positive.
The exponential-moment condition in Theorem 4.5 enters only through the control of the marked Poisson shot noise; a natural testable extension is whether a finite $(2d-1+\epsilon)$-th moment of $\rho$ would suffice if the interference estimate is handled by truncation instead of the moment-generating function.
The subcritical criterion $\alpha\beta > 2d-1$ suggests a phase diagram in the two exponents: for fixed dimension, percolation at low density is governed only by the product of the power-tail exponent and the path-loss exponent, not by either alone. Plotting or simulating the critical intensity against $\alpha$ and $\beta$ would test whether the product form is sharp.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: Chapter 4 is the reason to look at this. Loeffler weakens Kong-Yeh's bounded-power assumption for SINR percolation on PPPs to nonnegative integrable powers, treats d≥2, and gives a subcritical phase under tail decay. The subcritical proof is a sound reduction to Meester-Roy: connections in the SNR graph are contained in a Boolean model with random radii l^{-1}(N0τ/ρ), and the moment computation with αβ>2d−1 checks out. The expository chapters on the Boolean model and on Tóbiás's Cox SINR work are also solid. The supercritical proof, Theorem 4.5, is where the trouble is. The lattice construction has a boundary problem: Definition 4.6 sets δ = l^{-1}(N0τ/r)/2, and if r equals N0τ/l(0), then δ=0 and the good-site condition collapses. That alone is fixable by requiring r>N0τ/l(0). But Step 3 has a deeper gap. Good sites guarantee a path in the Gilbert graph gδ between points Xi∈Q1(z) and Xj∈Q1(z'); they do not imply |Xi−Xj|≤δ. A path of δ-edges can have long total length. And even if one replaces that false implication by the actual path, the intermediate vertices are arbitrary Poisson points; their powers need not exceed r, so the SINR lower bound, which needs transmitter power >r, does not apply. The proof gives no route through low-power relays. This is the load-bearing transfer from lattice percolation to SINR percolation, and as written Theorem 4.5 is not established. I do not read this as dishonesty. The author states conditions carefully, attributes building blocks correctly (Meester–Roy, Tóbiás), and the subcritical half is genuinely correct. But the supercritical claim needs serious repair: define good sites via a percolating network of high-power points, or prove a path whose every transmitter has power >r, and handle the boundary case r=N0τ/l(0). The typos—ε rather than r in the final displayed calculation, and the wrong inequality direction in Proposition 4.8—are minor by comparison. Who should read it: people working in continuum percolation for interference-limited networks, and anyone supervising a thesis student through a first original proof. It deserves a serious referee, not a desk reject; with the Step 3 gap fixed it would be a serviceable extension of Kong–Yeh. I would accept it for review but ask for a rewritten supercritical section.

Referee Report

3 major / 5 minor

Summary. This thesis studies continuum percolation in three multihop wireless-network models. Chapters 2 and 3 are expository, covering the Boolean model following Meester–Roy and the SINR model for Cox point processes following Tóbiás. The original contribution is Chapter 4, which introduces the SINR model with random powers on a Poisson point process and claims a phase transition: a subcritical phase when the path-loss function decays as l(r) ≤ r^{-β} and the power tail satisfies P(ρ>r) ≤ r^{-α} with αβ > 2d−1 (Corollary 4.4), and a supercritical phase when P(ρ>r)>0 for some r ≥ N0τ/l(0) and E[e^{αρ}]<∞ (Theorem 4.5). The subcritical proof reduces to the Boolean model with random radii, while the supercritical proof follows the three-step renormalization scheme of Dousse et al. The paper is written as a Master's thesis, with large parts devoted to background material.

Significance. If Theorem 4.5 and Corollary 4.4 are established, the paper would generalize the bounded-power results of Kong and Yeh to unbounded random powers having a positive probability of exceeding a threshold, in dimensions d≥2; the subcritical condition is natural and not previously stated in this generality. The expository parts are standard but well organized, and the subcritical part of Chapter 4 appears sound. However, as detailed below, the supercritical proof of Theorem 4.5 contains a load-bearing gap, so the main positive claim is not yet established as written. The paper's machine-checkable proofs are not present; the arguments are conventional pen-and-paper proofs with citations to external theorems (Meester–Roy, Tóbiás, Dousse et al.), and the subcritical reduction uses those theorems correctly.

major comments (3)

[Section 4.3, Step 3 (after Definition 4.6)] The proof asserts that if two high-power points Xi∈Q1(z) and Xj∈Q1(z') are connected by a path in gδ(Xλ)∩Q6(z), then |Xi−Xj|≤δ. This implication is false: a path consisting of edges of length at most δ can have endpoints arbitrarily far apart within Q6(z). Moreover, the intermediate vertices along such a path are not known to have power >r, so the SINR lower bound (which requires transmitter power greater than r) cannot be applied to every edge of the path. Consequently, percolation of the lattice does not imply an infinite path in the SINR graph as currently argued, and Theorem 4.5 is not established. The transfer argument needs to be reworked, for example by constructing connectivity through the marked high-power process rather than through the full unmarked Poisson process.
[Section 4.3, Theorem 4.5(a) and Definition 4.6] The theorem assumes r ≥ N0τ/l(0), but the proof requires the strict inequality r > N0τ/l(0). Indeed, δ is defined by δ = l^{-1}(N0τ/r)/2; at r = N0τ/l(0), strict decrease of l gives δ=0, the graph gδ has no edges between distinct Poisson points almost surely, and condition (2) of Definition 4.6 cannot hold for large λ. In that case Proposition 4.8 fails. The theorem statement should assume r > N0τ/l(0), and the presentation should note that the equality case is excluded.
[Section 4.3, Proposition 4.8 and Definition 4.6] The proof asserts lim_{λ→∞} P(A(o)=0)=0 without a quantitative argument, and the definition of 'good' connects all points of Xλ inside Q3(z), including low-power points. Even if the limit statement were proved, it would not provide the high-power connectivity needed in Step 3, because the SINR transfer requires a path whose vertices all have power above r. This is closely related to the first comment, but it indicates that the renormalization step must be reformulated at the level of the marked high-power process; the current Definition 4.6 is not sufficient for the claimed conclusion.

minor comments (5)

[Section 4.3, Step 3] In the SINR inequality, the symbol ϵ appears where r is meant: 'ρi l(...) > ϵ l(δ)' should read 'ρi l(...) > r l(δ)'.
[Section 4.3, Proposition 4.8] The phrase 'at least 7 d distant' is ambiguous; the independence argument should specify a numerical separation such as 13^d or 7^d and justify the lower bound m ≥ N/7^d on the size of the independent subset.
[Chapter 4 model definition] The conditions on the path-loss function in Chapter 4 (continuous, strictly decreasing on supp l, 1≥l(0), integrable) differ from those in Chapter 3, where additionally l(0)>τN0/P is assumed; this distinction should be stated explicitly so that the reader understands why rB is not used in Chapter 4.
[Abstract and Introduction] The abstract says that a supercritical phase exists if the random power has a 'large enough essential supremum', but condition (a) of Theorem 4.5 is a probability condition P(ρ>r)>0; the wording should be aligned with the theorem.
[Throughout] There are numerous typos and grammatical errors (for example, 'ﬁnitly', 'proced', and the equation numbering in Chapter 3 is occasionally inconsistent); the manuscript needs a careful language and formatting pass before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the random-powers SINR results are derived as sufficient conditions from external percolation theorems, with no input-output equivalence.

full rationale

The thesis's Chapter 4 does not fit its conclusion into its hypotheses. Theorem 4.5 assumes only a positive-probability power threshold and an exponential moment; the proof builds an auxiliary lattice whose open sites require locally high power, Gilbert-graph connectivity, and bounded interference. These are sufficient conditions, not restatements of SINR percolation. The subcritical phase (Proposition 4.1, Corollary 4.4) is obtained by bounding the SNR graph by a Boolean model with random radii and invoking Meester-Roy (Theorem 2.16), an external result. The supercritical lattice step uses external stabilization/renormalization results and Tobias's lattice-sum bound; Tobias is not an author of this thesis, and the bound is a parameter-free estimate, not the target theorem. No parameter is fitted to data and renamed a prediction. The proof does contain a genuine non-circular correctness gap: in Step 3, a path in g_delta has many edges and does not imply |Xi-Xj| <= delta, and intermediate relays need not have power > r; this undermines Theorem 4.5 as written, but it is a mathematical error, not a self-referential reduction. Hence circularity score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard point process theory (Campbell's theorem, Laplace functionals, the Marking Theorem), on the dependent-percolation result of Liggett, Schonmann and Stacey, and on a uniform lattice-sum estimate for the shifted path-loss function cited from Tobiás. It also relies on the standard fact that a Gilbert graph on a Poisson point process in a bounded cube becomes connected as the intensity tends to infinity. No fitted parameters or invented entities appear; all constants in the proofs are constructed explicitly.

assumptions (4)

standard math Campbell's theorem, Laplace functionals, and the Marking Theorem for Poisson and marked Poisson point processes.
Used in Proposition 4.9 to compute the Laplace transform of the shot noise process and to represent the marked PPP as a PPP on the product space.
standard math Dependent percolation domination by product measures (Liggett, Schonmann, Stacey, 1997).
Used to conclude lattice percolation from multi-site decay estimates in Chapter 3 and implicitly in Chapter 4.
standard math Uniform lattice-sum bound: for the shifted path-loss function l_6, there is a constant K0 < infinity such that Sigma_{z in Z^d} l_6(|z-x|) ≤ K0 for all x.
Cited from Tobiás and used in Proposition 4.9 to control the exponential moment of the interference; it follows from integrability of l, but is not proved in the thesis.
standard math Connectivity of the Gilbert graph on a Poisson point process in a fixed cube tends to 1 as the intensity tends to infinity, for any positive connection radius.
Used in Proposition 4.8 to show P(A(o)=0) → 0 as lambda → infinity; a standard random-geometric-graph fact not stated or cited.

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Pith. "Pith review of Percolation phase transitions for the SIR model with random powers." pith.science (2026). https://pith.science/paper/BKZHW4EO

@misc{pith2026190807375,
  author       = {Pith},
  title        = {Pith review of: Percolation phase transitions for the SIR model with random powers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BKZHW4EO}},
  note         = {Machine review of arXiv:1908.07375}
}

abstract

This thesis considers three models which describe a multihop ad-hoc telecommunication system. These systems consist of users sending messages, which can jump to other users to reach the target user. The first two models have already been examined extensively, whereas it is the first time the third model is studied. In all these models our goal is to understand under which conditions users situated far away from each other can communicate. This is the fundamental question of continuum percolation, which was introduced by Gilbert (1961), who established the Boolean model for Poisson point processes. In the first part of this thesis we introduce this model and prove the existence of a phase transition between a subcritical phase, where no infinite connected component of users exists, and a supercritical phase, where such a component exists. We also consider the case of random connection radii, following Meester and Roy (1996). In the second part we study the SINR model for Cox point processes in two or higher dimensions, following T\'obi\'as (2019). He proved that there exists a phase transition under certain stabilization and connectedness conditions on the intensity measure. The SINR model for homogeneous Poisson point processes in $\mathbb R^2$ was introduced by Dousse et al. (2005). In the third part we study percolation in the SINR model with random powers for Poisson point processes in $\mathbb R^d$, $d \geq 2$. This model was studied by Kong and Yeh (2007) for $d=2$ under strong boundedness conditions on the random power. In this thesis we weaken these conditions and only assume that the random power is nonnegative, integrable and not a.s. zero. We show that there exists a subcritical phase if the path-loss function decays fast enough. Further, we prove that a supercritical phase exists if random power has a large enough essential supremum and some exponential moments.

Figures

Figures reproduced from arXiv: 1908.07375 by the authors.

**Figure 2.** Realization of a MG (left) and a corresponding NMG (right) [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗

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Works this paper leans on

19 extracted references · 19 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

and Blaszczyszyn, B

Baccelli, F. and Blaszczyszyn, B. , Stochastic Geometry and Wireless Networks: Volume I: Theory, Now Publishers Inc., 2009

work page 2009

[2] [2]

Connectivity in Sub-Poisson Networks

Blaszczyszyn, B. and Yogeshwaran, D. , Connectivity in Sub-Poisson Networks , Proc. of 48th Annual Allerton Conference, see also: arXiv:1009.5696 , 2010

work page Pith review arXiv 2010

[3] [3]

and Yogeshwaran, D

Blaszczyszyn, B. and Yogeshwaran, D. , Clustering and percolation of point processes , Electron. J. Probab. 18, 1--20 , 2013

work page 2013

[4] [4]

and Baccelli, F

Dousse, O. and Baccelli, F. and Thiran, P. Impact of Interferences on Connectivity in Ad Hoc Networks , IEEE/ACM Trans. Networking 1, 425--436 , 2005

work page 2005

[5] [5]

and Franceschetti, M

Dousse, O. and Franceschetti, M. and Macris, N. and Meester, R. and Thiran, P. , Percolation in the Signal to Interference Ratio Graph , J. Appl. Prob. 43, 552--562 , 2006

work page 2006

[6] [6]

and Meester, R

Franceschetti, M. and Meester, R. , Random Networks for Communication -- From Statistical Physics to Information Systems, Cambridge University Press , 2007

work page 2007

[7] [7]

Random plane networks , Journal of the Society for Industrial and Appl

Gilbert, E.N. Random plane networks , Journal of the Society for Industrial and Appl. Math. 9, 533--543 , 1961

work page 1961

[8] [8]

and Khrisnapur, M

Ghosh, S. and Khrisnapur, M. and Peres, Y. , Continuum percolation for G aussian zeroes and G inibre eigenvalues , Ann. Probab., 44:5 , 2016

work page 2016