Percolation for the Finitary Random interlacements

Eviatar B. Procaccia; Jiayan Ye; Yuan Zhang

REVIEW 2 major objections 4 minor 25 references

Percolation for the Finitary Random interlacements

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Finitary random interlacements on $\mathbb{Z}^d$ undergo a connectivity phase transition driven by the average killed-walk length, with a unique infinite cluster for large $T$ and none for small $T$.

desk verdict New phase transition result for finitary random interlacements on Z^d, but the proof only establishes the supercritical side for T=R^3; the extension to all large T needs an unproved uniformity. read the letter →

arxiv 1908.01954 v2 pith:YQQHZVIV submitted 2019-08-06 math.PR

Eviatar B. Procaccia , Jiayan Ye , Yuan Zhang This is my paper · ORCID

classification math.PR MSC 60K3582B4360G55

keywords finitaryrandominterlacementspercolationphasetransitionuniqueinfiniteclusterZ^dlatticekilledwalkrenormalizationblockconstructioncontour-countingsubcriticalbound

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 proves that finitary random interlacements $\mathcal{FI}^{u,T}$ on $\mathbb{Z}^d$ ($d\ge 3$) undergo a phase transition in the average stopping time $T$: for every $u>0$, the occupied set has no infinite connected component for all sufficiently small $T$, and a unique infinite connected component for all sufficiently large $T$, almost surely. The result answers, in the special case of $\mathbb{Z}^d$, a question posed in the paper that introduced finitary random interlacements. The supercritical direction is obtained by coupling the finite killed walks to random interlacements, proving that large boxes are 'good' with high probability when $T=R^3$, and then showing the good-box field stochastically dominates supercritical site percolation. The subcritical direction uses a contour-counting estimate with a killed-capacity bound.

What carries the argument

The central object is the good-box event $\hat B(R)$ of Definition 3: a large box is good when, in every one of $O(R)$ subboxes, the first independent copy of the FRI produced a connected cluster of capacity at least $R^{2(d-2)/3}$, neighbouring such clusters are connected by the second independent copy, and no starting vertex far away sends a killed walk across the box. The argument's engine is a coupling of the finite killed walks with random interlacements (Section 4.1): trajectories of FRI that survive at least $T_0$ steps and whose backward parts avoid the set are shown to dominate truncated interlacement trajectories at level $uq$ with $q>1/2$, reducing the estimates to known capacity results. The renormalization step then treats the $9$-dependent field of good boxes as supercritical site percolation via a domination-by-product-measures theorem.

What would settle it

Evaluate the probability that the large box $\hat B(R)$ is good in $\mathcal{FI}^{u,T}$ at $T=(R+1)^3$ as $R$ grows; if for some $u>0$ this probability does not tend to $1$, the step from the subsequential result to Theorem 1 collapses and only the subsequence $T=R^3$ is established.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that connectivity of finitary random interlacements $\mathcal{FI}^{u,T}$ in $\mathbb{Z}^d$, $d\ge 3$, is governed by $T$: there exist $0<T_0(u,d)\le T_1(u,d)<\infty$ such that for $0<T<T_0$ the occupied set has no infinite connected component almost surely, while for $T>T_1$ it has a unique infinite connected component almost surely. The supercritical half is proved by showing that for $T=R^3$, the probability that a large box is good tends to $1$ as $R\to\infty$ (Theorem 3); this makes the collection of good boxes dominate an independent supercritical site percolation, giving an infinite cluster (Corollary 5.1), and ergodicity together with a classical uniqueness argument for percolation forces uniqueness (Theorem 4). The subcritical half is proved by a contour-counting estimate showing that the expected number of open self-avoiding paths of length $n$ from the origin decays exponentially for small $T$.

Load-bearing premise

The proof of the supercritical phase for every sufficiently large $T$, rather than only for $T=R^3$, rests on the unstated assumption that the Section 4 probability estimates hold uniformly for all $T$ between $R^3$ and $(R+1)^3$.

Editorial extensions

If this is right

For each $u>0$ and $d\ge 3$, the model has a definite regime of large $T$ in which $\mathcal{FI}^{u,T}$ almost surely contains exactly one infinite connected component, and a regime of small $T$ in which it almost surely contains none.
Because Theorem 1 holds for both bond and site percolation while Theorem 2 is proved for the edge-crossing convention, the supercritical existence of a unique infinite cluster is robust to the choice of connectivity.
For all sufficiently large integer $R$, $\mathcal{FI}^{u,R^3}$ almost surely has a unique infinite cluster, so the phase transition is realised along a natural subsequence of $T$.
Along with the known weak* convergence of FRI to random interlacements as $T\to\infty$, the result makes precise the sense in which the long-walk limit of FRI is percolative, matching the connectedness of $\mathcal{I}^u$.

Reading between the lines

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

A direct way to close the gap in Theorem 1 is to prove that the constants in Lemmas 4.6–4.10 and the coupling of Section 4.1 are uniform in $T\in[R^3,(R+1)^3]$; the natural starting point is Lemma 4.8's Poisson parameter $2du/(T+1)$, which already varies with $T$.
The good-box construction might transfer to other amenable graphs of polynomial growth, where capacity bounds and invariance principles are available, giving the first non-$\mathbb{Z}^d$ cases of the question that motivated this paper.
Since the paper notes $\mathcal{FI}^{u,T}$ is not monotone in $T$, the true critical value, if unique, would require a different argument; numerical estimates of cluster connectivity across $T$ for $d=3$ could indicate whether $T_0=T_1$ is plausible.

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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

Here's my take on the FRI percolation paper. The core result is new and worth knowing: for finitary random interlacements on Z^d, d≥3, the authors prove a phase transition in the stopping time T, with uniqueness of the infinite cluster in the supercritical regime. That answers Bowen's question for the amenable lattice case, which only the non-amenable case had been settled. The subcritical Peierls argument is clean: they bound the expected number of open self-avoiding paths via killed capacity and get exponential decay once T is small. The supercritical side is a standard block construction, coupling FRI to random interlacements and importing the Rath–Sapozhnikov estimates. The use of [21] is explicit and legitimate.

The catch is that Theorem 1, as stated, is not proven. The entire supercritical machinery—Lemmas 4.6–4.10 and the stochastic domination in Section 5—is written for T=R^3. The final sentence of Section 5 says that for an arbitrary large T you take R=floor(T^{1/3}) and the proof is complete. But that requires all the Section 4 estimates to hold uniformly for T between R^3 and (R+1)^3, and no uniformity statement appears. The model is not monotone in T: increasing T lowers the number of walks but lengthens each one, so you cannot dominate the behavior at larger T by the cube case. Without uniformity, the theorem only establishes the subsequence T=R^3. This is not a minor presentational issue; it's a genuine gap in the stated result.

How much does this matter? For the existence of a phase transition along the subsequence, the paper is fine. For the "all sufficiently large T" claim, a referee would need the authors to either prove uniformity or revise the statement to T=R^3 (or otherwise provide a density argument, which doesn't work here because of non-monotonicity). I expect the gap is fixable, since the estimates appear to depend on R only through coarse polynomial scales, but I can't verify that from the text.

The paper is otherwise careful. The ergodicity and Burton–Keane uniqueness argument follows the standard template. The acknowledgment of the later Brownian percolation work is appropriately placed.

Verdict: this deserves a serious referee—the result is important within the interlacements subfield and the gap, while real, is plausibly repairable. It's not a desk reject. I would send it to review, with instructions that the rounding step be justified or the theorem statement weakened.

Referee Report

2 major / 4 minor

Summary. This paper studies the connectivity properties of finitary random interlacements FI^{u,T} on Z^d, d>=3. The main results are a subcritical statement (Theorem 2): for each u>0 and all sufficiently small T, FI^{u,T} has no infinite connected component almost surely, proved by a Peierls argument; and a supercritical statement (Theorem 1): for each u>0 and all sufficiently large T, FI^{u,T} has a unique infinite connected component almost surely, proved by a block renormalization argument. The supercritical proof is carried out for stopping times of the special form T=R^3: Theorem 3 and Lemma 5.1 establish that good boxes occur with high probability and that the induced block process dominates a supercritical site percolation, Corollary 5.1 gives existence of an infinite cluster for FI^{u,R^3}, and Theorem 4 gives uniqueness for FI^{u,R^3}. The passage from this subsequence to all sufficiently large T is made in the final paragraph of Section 5 by setting R=floor(T^{1/3}).

Significance. If the uniformity issue described below is resolved, the paper would answer Bowen's Question 2 in the special case of Z^d and would provide the first phase transition for finitary random interlacements. The subcritical Peierlis argument is self-contained and gives an explicit exponential decay of the cluster of the origin. The supercritical block construction is a careful adaptation of the Rath--Sapozhnikov renormalization strategy to the killed setting, and the paper is honest about the non-monotonicity of the model in T and about the open problems that remain. The proof relies on several nontrivial estimates imported from [21], which are used in a black-box manner. As written, however, the central supercritical theorem is only proved for the subsequence T=R^3, and the claimed extension to all sufficiently large T is not supported.

major comments (2)

[Section 5, final paragraph (after Corollary 5.1)] Theorem 1 as stated is not established. Corollary 5.1 and Theorem 4 prove existence and uniqueness of the infinite cluster for FI^{u,R^3} for large integer R. The final paragraph says that for sufficiently large T one takes R=floor(T^{1/3}) 'and the proof is complete'. This is not justified: all the estimates in Section 4 are formulated for T=R^3, for example Lemma 4.6 uses the geometric walk variable Y_{x,R^3}, Lemma 4.8 bounds the Poisson parameter by 2du/(R^3+1), Lemma 4.7 uses the R^{2.5} truncation, and Lemma 4.10 has the same R^3 dependence. No statement in the paper proves these estimates uniformly for T in [R^3,(R+1)^3). Since the model is not monotone in T, as the paper itself observes in Section 1.1, one cannot pass from the subsequence T=R^3 to all large T by domination. The theorem should either be weakened to the subsequence or supplied with an explicit uniformity lemma for the Section 4 estimates.
[Section 6, proof of Theorem 4, event E_n] In the uniqueness proof, the authors write 'By Lemma 4.8, the probability of event E_n^c decays stretch exponentially', where E_n is the event that no killed random walk starting in Z^d \ B(2n) intersects B(n). Lemma 4.8 is stated for FI^{u,R^3} with the box hat B(R) of radius 64R^2 and the starting set outside B(128R^2); it does not cover the regime of fixed T=R^3 and n tending to infinity. A separate estimate, for example using the diffusive time scale n^2 versus the mean lifetime R^3, is needed. This gap is local and likely fixable, but as written the cited lemma is not literally applicable.

minor comments (4)

[Section 4.3, Lemma 4.6] The claim that N^{(1)}_{4R,1} stochastically dominates hat N^{(1)}_{4R,1} is not literally true, because several successful pairs (x,i) can share the same vertex x while N^{(1)} counts vertices. The subsequent argument only needs the implication hat N^{(1)} >= 1 implies N^{(1)} >= 1, so the proof can be repaired by replacing 'stochastically dominates' with this implication or by defining hat N as the number of distinct vertices with at least one successful pair.
[Section 4.3, end of subsection] After the proof for the representative subbox b_{4R,1}, the passage to condition (1*) for all 0 <= i <= 8R and 1 <= j <= d should include an explicit union bound over the O(R) subboxes; the failure probability e^{-R^{1/18}} makes this immediate, but the union bound is not written.
[Sections 4.3 and 4.4] The imported estimates from [21] (Lemmas 6, 7, 8, 11 and 12) are used in a black-box way without precise statements. For reproducibility, please state the exact forms of these lemmas, or at least the precise inequalities and hypotheses that are being invoked.
[Section 7, equation (6)] The displayed bound e^{-t_0 n}((1-T)/(1-T e^{t_0}))^L <= (6d)^{-n} 2^{n+1} is not immediate from the definitions, because L is proportional to n+1 with a constant depending on u. The intended argument is that T_0 is chosen so that the base, raised to the constant power ceil(eu(2d)+log(3d)), is at most 2; this should be spelled out.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the phase-transition proof is built on external inputs (Bowen's FRI formalism and Rath–Sapozhnikov estimates), with no fitted parameter renamed as a prediction and no load-bearing self-citation chain.

full rationale

The paper's central claims in Theorems 1 and 2 are derived from explicit probabilistic estimates and couplings, not from assuming the conclusion. The supercritical direction is proved by defining good boxes, proving with high probability that a fixed box is good at T=R^3 (Theorem 3, Lemmas 4.6–4.10), and then using a renormalization argument to obtain an infinite cluster, with uniqueness via ergodicity and a Burton–Keane-style argument. The subcritical direction uses a Peierls argument with explicit Poisson and geometric tail bounds. The key ingredients from the literature are external to the present authors: Bowen's construction of FRI and convergence results, and Rath–Sapozhnikov's estimates on random interlacements. These are not self-citations that smuggle in the target result. The final step of Theorem 1, setting R=floor(T^{1/3}) for general large T, is not shown to preserve the uniformity of the R^3 estimates, and the model is not monotone in T; however, this is a possible correctness gap or missing uniformity argument, not a circular reduction. No equation in the paper rewrites the claimed theorem as an assumption of itself, and no fitted quantity is later presented as a prediction. Accordingly, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof is a standard probabilistic existence proof: no empirical parameters are fitted, and the only mathematical input beyond standard potential theory is Bowen's FRI restriction theory and the Rath-Sapozhnikov estimates for random interlacements. The one unproved premise is the uniformity over T near R^3, listed as an ad hoc axiom.

assumptions (6)

standard math Capacity bounds for boxes and monotonicity of capacity (Lemmas 4.4-4.5 of the paper, from Lawler-Limic and Drewitz et al.).
Used to convert Poisson counts of boundary trajectories into high-probability lower bounds for the good box event; standard potential theory for transient random walk on Z^d, d>=3.
domain assumption FRI restriction to a finite set is a Poisson point process with intensity u*cap^(T)(K) (Proposition 2.2, Lemma 2.1, Corollary 2.1, taken from Bowen [2] and Poisson process theory).
This is the structural backbone: Lemma 2.2 lets the authors sample trajectories hitting a box as i.i.d. killed walks with the killed equilibrium measure, and the subcritical proof counts N_gamma as Poisson with parameter u*cap^(T)(gamma). If this failed, both phases would be unproved.
domain assumption The Rath-Sapozhnikov estimates for random interlacements (Lemmas 6, 7, 8, 11, 12 of [21]) transfer to finitary interlacements through the coupling of Section 4.1.
The good-box probability and the crossing probability inequality (4) rest on these imported estimates; the transfer assumes the truncated FRI trajectories faithfully approximate random interlacements at scales R^{1.6} and R^{2.5}.
ad hoc to paper All Section 4 estimates hold uniformly for T in [R^3, (R+1)^3).
This is required to pass from the proved T=R^3 case to 'all sufficiently large T' in Theorem 1; the paper asserts the passage without proof in the last paragraph of Section 5.
standard math Liggett-Schonmann-Stacey stochastic domination by product measures applies to the 9-dependent good-box field.
Used in Lemma 5.1 to show that a high density of good boxes implies an i.i.d. supercritical site percolation and hence an infinite cluster.
standard math Burton-Keane uniqueness machinery (finite energy, translation invariance, ergodicity) applies to FRI.
The uniqueness proof in Section 6 adapts Burton-Keane and its trifurcation counting argument; it relies on the FRI lattice-shift ergodicity proved in Proposition 6.1 and on the possibility of resampling local trajectories with positive probability.

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Pith. "Pith review of Percolation for the Finitary Random interlacements." pith.science (2026). https://pith.science/paper/YQQHZVIV

@misc{pith2026190801954,
  author       = {Pith},
  title        = {Pith review of: Percolation for the Finitary Random interlacements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YQQHZVIV}},
  note         = {Machine review of arXiv:1908.01954}
}

abstract

In this paper, we prove a phase transition in the connectivity of Finitary Random interlacements $\mathcal{FI}^{u,T}$ in $\mathbb{Z}^d$, with respect to the average stopping time. For each $u>0$, with probability one $\mathcal{FI}^{u,T}$ has no infinite connected component for all sufficiently small $T>0$, and a unique infinite connected component for all sufficiently large $T<\infty$. This answers a question of Bowen in the special case of $\mathbb{Z}^d$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

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

[1] [21]

R´ ath and A

B. R´ ath and A. Sapozhnikov. On the transience of random interlacements. Electronic Communica- tions in Probability , 16:379–391, 2011

work page 2011

[2] [1]

Biskup and E

M. Biskup and E. B. Procaccia. Eigenvalue versus perimet er in a shape theorem for self-interacting random walks. Ann. Appl. Probab. , 28(1):340–377, 2018

work page 2018

[3] [2]

L. Bowen. Finitary random interlacements and the gabori au-lyons problem. arXiv preprint arXiv:1707.09573, 2017

work page Pith review arXiv 2017

[4] [3]

R. M. Burton and M. Keane. Density and uniqueness in perco lation. Communications in Mathe- matical Physics , 121(3):501–505, 1989

work page 1989

[5] [4]

ˇCern` y and S

J. ˇCern` y and S. Popov. On the internal distance in the interlac ement set. Electronic Journal of Probability, 17, 2012. 21

work page 2012

[6] [5]

Drewitz, B

A. Drewitz, B. R´ ath, and A. Sapozhnikov. An introduction to random interlacements . Springer, 2014

work page 2014

[7] [6]

Drewitz, B

A. Drewitz, B. R´ ath, and A. Sapozhnikov. On chemical dis tances and shape theorems in percolation models with long-range correlations. Journal of Mathematical Physics , 55(8):083307, 2014

work page 2014

[8] [7]

Durrett and D

R. Durrett and D. Griﬀeath. Supercritical contact proce sses on Z. Ann. Probab., 11(1):1–15, 1983

work page 1983