{"id":"4528622d-2a94-4a61-94d9-0493d1345c6c","arxiv_id":"1908.01954","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every u>0, finitary random interlacements on Z^d with d>=3 have no infinite component for small mean walk length T and a unique infinite component for large T.","lead":"Finitary random interlacements are random networks built from many short random walks on a high-dimensional grid; this paper proves that as the average walk length grows, the network switches from having no giant connected component to having exactly one. It answers an open question of Bowen for the integer lattice Z^d.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 as stated ('all sufficiently large T') is not established: the supercritical proof is written only for T=R^3, and the final rounding step R=floor(T^{1/3}) requires an unproved uniformity of Section 4 estimates.","rationale":"The reader's weakest_assumption identifies exactly the step I find most load-bearing: the passage from the subsequence T=R^3 to all sufficiently large T in Theorem 1. Since the model is non-monotone in T, as stated in Section 1.1, this is not a trivial extension. My read of the remaining components: the subcritical Peierls argument in Section 7 appears coherent; the uniqueness argument in Section 6 is a standard Burton-Keane adaptation with positive-probability resampling events; the imported interlacement estimates from [21] are cited and not in dispute here. The only substantive gap is the unproved uniformity of Section 4 estimates over T in [R^3,(R+1)^3]. The gap is plausibly fixable because T/R^3 tends to 1 in the relevant range, but the manuscript never supplies the verification, so the theorem as stated is not fully proved. I therefore recommend CONDITIONAL, with acceptance conditioned on supplying the uniformity argument or on weakening Theorem 1 to the subsequence T=R^3.","tokens_in":18570,"tokens_out":12260,"duration_ms":177771,"concrete_test":"Re-derive the estimates in Section 4 with T=(R+1)^3, or more generally T=R^3(1+o(1)), instead of T=R^3. Concretely: (i) check q1(R)=P(Geom(1/(T+1)) >= R^{1.6}) > 1/2 uniformly; (ii) check the union bound in Lemma 4.8 still tends to 0 when the Poisson parameter is 2du/(T+1) and the killing survival rate is (T/(T+1))^{mR^{7/2}}; (iii) repeat for Lemma 4.10. If all three hold, Theorem 1 as stated follows and no weakening is needed. If any estimate fails, identify the first failure; then the correct theorem is only for FI^{u,R^3} with large R, and Theorem 1 must be restricted to a subsequence unless an additional argument is supplied.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 5 proves Corollary 5.1 only for FI^{u,R^3}; the proof of Theorem 1 then says that for arbitrary large T one takes R=floor(T^{1/3}) 'and the proof is complete'. The entire supercritical construction is formulated for the stopping-time parameter T=R^3: Lemma 4.6 uses geometric walks with mean R^3 and q1(R)>1/2, Lemma 4.8 bounds the Poisson parameter by 2du/(R^3+1), Lemma 4.10 has the same R^3 dependence, and the q=1/2 comparison in Section 4.4 is made for T=R^3. For general T between R^3 and (R+1)^3 none of these statements is proved; the rounding step silently requires all Section 4 estimates to hold uniformly on [R^3,(R+1)^3]. This is not a purely cosmetic point, because the model is not monotone in T: increasing T simultaneously lowers the number of starting walks and lengthens each walk, so one cannot pass from T=R^3 to larger T by domination. If the uniformity fails, Theorem 1 is only known for the subsequence T=R^3.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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}).","tokens_in":18801,"tokens_out":28348,"duration_ms":287625,"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":[{"comment":"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":"Section 5, final paragraph (after Corollary 5.1)"},{"comment":"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.","section":"Section 6, proof of Theorem 4, event E_n"}],"minor_comments":[{"comment":"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":"Section 4.3, Lemma 4.6"},{"comment":"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.","section":"Section 4.3, end of subsection"},{"comment":"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":"Sections 4.3 and 4.4"},{"comment":"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.","section":"Section 7, equation (6)"}],"recommendation":"major_revision","confidential_remarks":"The main gap in the paper is the passage from the subsequence T=R^3 to all sufficiently large T in Theorem 1. This is likely fixable by adding a uniformity statement for the Section 4 estimates over T in [R^3,(R+1)^3], since the parameter changes only by O(1/R^3) and the estimates appear to be continuous in T. If the authors cannot supply such a uniformity proof, Theorem 1 should be weakened to the subsequence T=R^3, which would still be a substantial result. The heavy reliance on [21] without stating the imported lemmas makes verification harder than necessary."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nHow 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.\n\nThe 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.\n\nVerdict: 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.","headline":"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.","tokens_in":19344,"tokens_out":3118,"would_cite":true,"duration_ms":32687,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B43","60G55"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $T$.","keywords":["finitary random interlacements","random interlacements","percolation phase transition","unique infinite cluster","Z^d lattice","killed random walk","renormalization block construction","contour-counting subcritical bound"],"falsifier":"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.","tokens_in":18339,"feed_emoji":"🕸️","tokens_out":9922,"duration_ms":90078,"temperature":0.7,"pith_summary":"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.","feed_headline":"Killed random walks percolate when long enough","feed_subtitle":"Small mean walk length leaves no infinite component; large mean length gives exactly one.","key_machinery":"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.","core_discovery":"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 $T$.","pith_inferences":["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."],"forward_implications":["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 $\\mathcal{I}^u$."],"supporting_citations":[{"why":"Introduces finitary random interlacements, proves weak* convergence to random interlacements, and poses the question this paper answers for $\\mathbb{Z}^d$.","marker":"[2]"},{"why":"Provides the capacity estimates used to show good boxes occur with high probability under the FRI–RI coupling.","marker":"[21]"},{"why":"Defines random interlacements and proves the ergodicity used in the coupling and uniqueness sections.","marker":"[22]"},{"why":"Supplies the Poisson point process and killed capacity facts that represent FRI locally and bound path probabilities.","marker":"[5]"},{"why":"Gives the domination-by-product-measures theorem that converts the 9-dependent good-box field into a supercritical iid site percolation.","marker":"[15]"},{"why":"Provides the classical argument, adapted in Section 6, that rules out two or infinitely many infinite clusters.","marker":"[3]"},{"why":"Supplies the contour-counting estimate adapted in Section 7 to bound the probability that a long self-avoiding path is open.","marker":"[16]"},{"why":"Gives the capacity bounds for boxes needed in Lemma 4.4 and Lemma 4.6.","marker":"[14]"}],"fun_headline_variants":["Finitary interlacements percolate when T is large","No infinite component for small stop times, one for large","Interlacements: T drives connectivity transition","Long killing times yield unique infinite cluster","Phase transition in finitary random interlacements"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $(R+1)^3$.","fun_headline_variants_meta":{"raw":{"variants":["Finitary interlacements percolate when T is large","No infinite component for small stop times, one for large","Interlacements: T drives connectivity transition","Long killing times yield unique infinite cluster","Phase transition in finitary random interlacements"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000699,"raw_usage":{"total_tokens":3106,"prompt_tokens":844,"completion_tokens":2262,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":460,"completion_tokens_details":{"reasoning_tokens":2189}},"tokens_in":460,"tokens_out":2262,"duration_ms":19053,"temperature":1.0,"reasoning_tokens":2189,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:00:26.325667+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Finitary random interlacements and the Gaboriau-Lyons problem","cited_arxiv_id":"1707.09573","evidence_quote":"Introduces finitary random interlacements, proves weak* convergence to random interlacements, and poses the question this paper answers for $\\mathbb{Z}^d$."},{"cited_title":"R´ ath and A","cited_arxiv_id":null,"evidence_quote":"Provides the capacity estimates used to show good boxes occur with high probability under the FRI–RI coupling."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines random interlacements and proves the ergodicity used in the coupling and uniqueness sections."},{"cited_title":"Drewitz, B","cited_arxiv_id":null,"evidence_quote":"Supplies the Poisson point process and killed capacity facts that represent FRI locally and bound path probabilities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the domination-by-product-measures theorem that converts the 9-dependent good-box field into a supercritical iid site percolation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical argument, adapted in Section 6, that rules out two or infinitely many infinite clusters."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the contour-counting estimate adapted in Section 7 to bound the probability that a long self-avoiding path is open."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the capacity bounds for boxes needed in Lemma 4.4 and Lemma 4.6."}],"review_version":1}