{"id":"58fb3df5-32f2-4a40-ab27-b1e0c4a8df91","arxiv_id":"2504.21378","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Effective resistances in one-dimensional critical long-range percolation grow as n^{δ(β)} for some δ(β) in (0,1).","lead":"This paper proves that in a one-dimensional random network where long-range connections decay like distance squared, the effective resistance to distance n grows as a power n^δ, with a nontrivial exponent δ between 0 and 1. The result gives the sharp growth rate of a quantity that controls random walks on such networks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The lower-tail proof of Proposition 5.8 assumes independence of annulus resistances that share a boundary annulus, causing positive correlation and invalidating the recursion.","rationale":"The reader's weakest_assumption concerns imported results ([11] and [3]) and omitted details in Proposition 6.1. My review identifies a more specific and more load-bearing internal gap: Proposition 5.8, which is needed for the box-to-box comparison in Theorem 1.1 and for the very-good interval estimates in Proposition 6.3 that underpin supermultiplicativity, contains a false independence claim. The resistances R_k and R_{k+2} in the lower-tail proof share an entire boundary annulus, so they are positively correlated by FKG; the product bound used to derive the recursion (5.40) is therefore invalid. This is not merely a missing detail but a concrete erroneous step. The main point-to-box part of Theorem 1.1 may still be salvageable, but the box-to-box part and the full supermultiplicativity are not proven as written. Because the defect is likely fixable by a more careful stochastic domination or by considering disjoint annuli, the appropriate verdict remains conditional rather than rejection.","tokens_in":72856,"tokens_out":22145,"duration_ms":210177,"concrete_test":"Recompute the probability in Proposition 5.8 after (5.38) using the FKG lower bound P(R_k < x, R_{k+2} < x) ≥ P(R_k < x) P(R_{k+2} < x) for the shared annulus, and check whether the resulting bound on having two small resistances is O(M max_i P(R_i < x)) rather than O(M^2 max_i P(R_i < x)^2). If the recursion (5.40) no longer converges below ε/2, the lower tail of the box-to-box resistance in Proposition 5.8 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Proposition 5.8, after equation (5.38), the proof bounds the probability that at least two of the odd-k resistances R_k := R(A_{MN+kN,MN+(k-1)N}, A_{MN+(k+2)N,MN+(k+1)N}; E\\ C_{\\ge N}) are below γ^3 Λ(N) by M^2 max_i P(R_i < γ^3 Λ(N))^2, citing the 'independence of R(...) for all odd k'. This independence is false. For k and k+2, the target annulus of R_k, namely A_{MN+(k+2)N,MN+(k+1)N}, is exactly the source annulus of R_{k+2}; both resistances are decreasing functions of the edge set, so by the FKG inequality they are positively correlated: P(R_k < x, R_{k+2} < x) ≥ P(R_k < x) P(R_{k+2} < x). Thus the intersection probability can be on the order of the marginal, not its square. The subsequent recursion (5.40) and the derived lower tail for R([-MN,MN],[-2MN,2MN]^c | A_{MN}) therefore do not follow. Since Proposition 5.8 is used in Proposition 6.3 and hence in the supermultiplicativity Proposition 6.1, this gap threatens the second half of Theorem 1.1 as well.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies critical long-range percolation on Z with edge probability 1-exp(-β∫∫|u-v|^{-2}dudv) and unit conductances, and proves that the effective resistance from 0 to [-n,n]^c, as well as the conditioned box-to-box resistance from [-n,n] to [-2n,2n]^c, both grow like n^{δ(β)} for some δ(β)∈(0,1). The proof defines Λ(n) as the maximum expected point-to-point resistance in [0,n), establishes submultiplicativity (Prop. 2.1), a weak form of supermultiplicativity (Prop. 4.1), a second-moment bound (Prop. 5.1), comparability of point-to-point, point-to-box, and box-to-box resistances (Props. 5.4 and 5.8), and finally full supermultiplicativity (Prop. 6.1). The exponent δ is then obtained via Fekete's lemma. The lower endpoint δ>0 is imported from a strengthened lower-bound lemma (Lemma 3.3) based on [11], and the upper endpoint δ<1 is imported from [3].","tokens_in":73134,"tokens_out":7283,"duration_ms":72851,"significance":"If the main theorem holds, this is a substantial result: it gives the sharp polynomial growth exponent for effective resistances in the critical one-dimensional long-range percolation model, a quantity of direct relevance to random walks. The paper's architecture—defining Λ(n), proving sub- and supermultiplicativity, and then comparing several resistance types—is natural and is supported by several nontrivial technical contributions, notably the flow-comparison Lemma 2.2, the detailed submultiplicativity proof, and the second-moment bound for point-to-point resistances. The exposition is generally careful and well organized. However, the proof of the box-to-box lower tail in Proposition 5.8 contains a false independence assertion that is load-bearing for the rest of the paper, and Proposition 6.1 is only sketched. These gaps need to be addressed before the main claims can be considered established.","major_comments":[{"comment":"The proof asserts 'independence of R(...) for all odd k' and uses it to bound the probability that at least two of the annulus resistances R_k are below γ^3Λ(N) by M^2 max_i P(R_i < γ^3Λ(N))^2. This independence is false: for k and k+2, the target annulus of R_k is exactly the source annulus of R_{k+2}, and both resistances are decreasing functions of the common edge set E\\ C_{\\ge N}. By FKG, P(R_k < x, R_{k+2} < x) ≥ P(R_k < x)P(R_{k+2} < x), so the intersection probability can be of order the marginal rather than its square. Consequently the recursion leading to (5.40) and the lower-tail bound for R([-MN,MN],[-2MN,2MN]^c | A_{MN}) do not follow. Since Proposition 5.8 is used in Proposition 6.3 and in the proof of the main theorem, this gap directly threatens Theorem 1.1.","section":"§5.3, Proposition 5.8, after Eq. (5.38)"},{"comment":"Proposition 6.1 is the second key ingredient in the proof of Theorem 1.1, but its proof is only a sketch: after stating the new very-good interval definition, the text says 'we will omit certain details' and then asserts the coarse-graining estimate (6.4), the properties (P1)-(P3), and inequality (4.46) without proof. The verification of these facts in the new setting is not a routine line-by-line copy of Section 4, because the definition of very-good intervals and the input Proposition 6.3 differ from Section 3. Since the supermultiplicativity of Λ(n) is essential for the existence of δ, this omission is load-bearing. The authors should either supply the full coarse-graining proof or state precisely which lemmas from Section 4 transfer unchanged and why.","section":"§6, Proposition 6.1"}],"minor_comments":[{"comment":"The notation E[·|bE] conditions on the edge set bE, but bE is not explicitly identified with a σ-algebra; please clarify that the conditional expectation is with respect to the σ-algebra generated by bE.","section":"§2.2, Eq. (2.19)"},{"comment":"There is an inconsistency in the constants: the text uses both e^{-3C4 m^{1/\\log n}} and e^{-3C3 m^{1/\\log n}} in the same proof. Please correct the notation.","section":"§3.3, proof of Proposition 3.2"},{"comment":"The parameters A, λ, a_k, b_k, b_{k-1,k}, and K_* are numerous and somewhat opaque; a short table or a summary of their roles would make the coarse-graining argument much easier to follow.","section":"§4.2, parameters in (4.10)-(4.13)"},{"comment":"The notation A^c_{2MN,MN} is used for the annulus [-2MN,2MN]\\[-MN,MN], but the superscript 'c' normally denotes complement; please use a distinct symbol or define the notation explicitly.","section":"§5.3, Definition 5.9"},{"comment":"There are several typographical issues: 'at leat' should be 'at least', and 'hods' in Section 5.1 should be 'holds'. Please proofread the final version.","section":"§5.3, proof of Proposition 5.8"},{"comment":"The application of Fekete's lemma to the sequences a_k and b_k is correct, but a sentence justifying finiteness of log Λ(2^k) (e.g., by the boundedness of the resistance of a single edge) would improve the exposition.","section":"§7, proof of Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The main mathematical concern is the incorrect independence claim in the proof of Proposition 5.8. If the authors can repair this step—for example by choosing a separated subsequence of annuli so that the relevant resistances are genuinely independent, or by a different second-moment estimate—the rest of the architecture appears sound. The proof of Proposition 6.1 should also be completed rather than sketched, and the status of Lemma 3.3 as a strengthening of [11] should be made more explicit. The paper is long but well organized, and the topic is appropriate for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a real new core: the comparison lemma for electric flows, the submultiplicativity proof, and the coarse-graining scheme adapted to resistance rather than graph distance. These are not imports from [3] and they hold up on inspection. The comparability between point-to-point, point-to-box, and box-to-box resistances is also a substantial step forward. I came away believing the upper bound is solid and the overall strategy is the right one.\n\nThe soft spot is where the reader landed, and it is worse than 'details omitted.' In the proof of Proposition 5.8, after equation (5.38), the paper asserts independence of the annulus resistances R(A_{MN+kN,MN+(k-1)N}, A_{MN+(k+2)N,MN+(k+1)N}; E\\setminus C_{\\ge N}) for odd k. For k and k+2, the target annulus of one is exactly the source annulus of the other. Both resistances are decreasing functions of the shared edge set inside that annulus, so the FKG inequality makes them positively correlated. The intersection probability is not the square of the marginal; it can be as large as the smaller marginal. The bound by M^2 max_i P(R_i < x)^2 is therefore unjustified. The recursion (5.40) and the lower tail for the box-to-box resistance do not follow. Since Proposition 5.8 feeds Proposition 6.3, which feeds Proposition 6.1, the second half of Theorem 1.1 is currently unsupported.\n\nA separate, smaller concern: Proposition 6.1 is explicitly sketched rather than proved in detail. That would be acceptable if the rest were airtight, but it is not.\n\nThe external inputs — [11] for the positive lower bound and [3] for δ<1 — are prior results, not restatements of the conclusion. The citation pattern is honest, and the paper is clearly written despite its length.\n\nWho should read this: anyone working on critical long-range percolation, effective resistance, or random walk on such clusters. The techniques in Sections 2–4 will be useful even if the current version is not the last word. But a serious referee should not pass it in this state. The dependence issue in Proposition 5.8 needs a genuine fix, possibly by decoupling the annuli through a gap or by a more careful martingale or negative-association argument. I would not cite the main theorem yet, but I would send it to a strong referee and ask for a major revision.","headline":"The upper bound and the comparison machinery are genuinely new, but the lower-tail recursion in Proposition 5.8 relies on a false independence claim that currently breaks the supermultiplicativity argument.","tokens_in":73675,"tokens_out":3966,"would_cite":false,"duration_ms":42521,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B27","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"Resistances grow as n^δ in critical long-range percolation","keywords":["long-range percolation","effective resistance","critical percolation","polynomial growth","random electric network","one-dimensional percolation","renormalization","multi-scale coarse graining"],"falsifier":"For a fixed $\\beta$ and large $n=2^k$, compute $\\mathbb{E}[R(0,[-n,n]^c)]$ (or estimate it by simulation over many samples) and check whether the ratio $\\mathbb{E}[R(0,[-n,n]^c)]/n^{\\delta}$ stays bounded away from $0$ and $\\infty$ as $k\\to\\infty$, where $\\delta$ is the slope of $\\log\\mathbb{E}[R(0,[-n,n]^c)]$ against $\\log n$. If that slope fails to converge to a value in $(0,1)$, or if the ratio diverges along a subsequence, Theorem 1.1 is false.","tokens_in":72646,"feed_emoji":"⚡","tokens_out":8745,"duration_ms":84020,"temperature":0.7,"pith_summary":"This paper proves that in the one-dimensional critical long-range percolation model, where an edge between $i$ and $j$ is present with probability roughly $\\beta|i-j|^{-2}$, the effective electrical resistance from the origin to the complement of $[-n,n]$ grows like $n^{\\delta(\\beta)}$ for a constant $\\delta(\\beta)\\in(0,1)$ depending only on $\\beta$. The same power law holds for the resistance between the intervals $[-n,n]$ and $[-2n,2n]^c$, conditioned on there being no edge directly joining them. A sympathetic reader should care because effective resistance is the natural metric for random walks on a random network, so this pins down how the critical model conducts current and, by extension, how a random walk spreads on it.","feed_headline":"Resistances grow as n^δ in critical long-range percolation","feed_subtitle":"Point-to-box and box-to-box resistances share one exponent δ(β), strictly between 0 and 1.","key_machinery":"The load-bearing object is $\\Lambda(n)=\\max_{i,j\\in[0,n)}\\mathbb{E}[R_{[0,n)}(i,j)]$, the largest expected point-to-point resistance inside an interval of length $n$ when edges leaving the interval are ignored. The argument shows several resistances of interest are comparable to $\\Lambda(n)$, and then proves that $\\Lambda$ is almost multiplicative: $\\Lambda(mn)\\asymp \\Lambda(m)\\Lambda(n)$ up to constants depending only on $\\beta$. Submultiplicativity comes from a renormalization of the line into blocks together with a comparison lemma stating that raising the conductance of one edge incident to a vertex cannot increase the current through any other incident edge. Supermultiplicativity is the harder direction: the paper introduces 'very good' intervals that force any unit flow to spend at least $\\alpha m^\\delta$ of energy, then uses a multi-scale coarse-graining argument to show that, with high probability, the rare bad intervals can be covered by 'good red animals' whose energy cost is negligible compared with the surrounding regions. Once $\\Lambda$ is multiplicative up to constants, a standard subadditivity limit gives $\\delta=\\lim_n \\log\\Lambda(n)/\\log n\\in(0,1)$.","core_discovery":"The central claim is Theorem 1.1: for every $\\beta>0$ there is an exponent $\\delta=\\delta(\\beta)\\in(0,1)$, depending only on $\\beta$, such that for all $n$ both $\\mathbb{E}[R(0,[-n,n]^c)]\\asymp_P R(0,[-n,n]^c)\\asymp_P n^\\delta$ and, conditionally on there being no edge between $[-n,n]$ and $[-2n,2n]^c$, $\\mathbb{E}[R([-n,n],[-2n,2n]^c)\\mid \\text{no such edge}]\\asymp_P R([-n,n],[-2n,2n]^c)\\asymp_P n^\\delta$. Here $\\asymp_P$ means that for every $\\varepsilon>0$ the ratio is bounded between constants with probability at least $1-\\varepsilon$. The proof obtains $\\delta$ by proving that the maximal expected point-to-point resistance $\\Lambda(n)$ inside an interval of length $n$ is both submultiplicative and supermultiplicative up to constants, so a standard subadditivity argument yields $\\delta=\\lim_n \\log\\Lambda(n)/\\log n$. The inequalities $0<\\delta$ and $\\delta<1$ are imported from earlier polynomial lower bounds and from the known chemical-distance exponent of the same critical model; the present contribution is showing that one well-defined exponent governs all these resistances and that point-to-box and conditioned box-to-box resistances are comparable to $\\Lambda(n)$.","pith_inferences":["A natural testable extension is whether $\\delta(\\beta)$ is strictly decreasing and, in particular, approaches $1$ as $\\beta\\to0$; the paper only records monotonicity, not limits.","The same coarse-graining machinery, with $\\Lambda(n)$ in place of the graph distance, may be adaptable to the two-dimensional critical model, where the analogous resistance exponent is not known; the paper does not make this claim.","Because the model is exactly scale-invariant, the theorem suggests that the resistance metric itself may admit a scaling limit with a Hausdorff-type exponent $\\delta(\\beta)$, even though the paper notes that deriving such a limit remains a major open challenge."],"forward_implications":["For every $\\beta>0$ the point-to-box resistance and the conditioned box-to-box resistance share one polynomial growth exponent $\\delta(\\beta)\\in(0,1)$, so no separate exponent is needed for different resistance types.","The lower-tail estimate of Corollary 1.2 holds: for sufficiently small $\\varepsilon>0$, $P(R(0,[-n,n]^c)\\ge\\varepsilon n^\\delta)\\ge 1-\\varepsilon^q$, so the resistance is bounded away from zero with overwhelming probability.","The paper's resistance estimates place the critical one-dimensional model in position to support random-walk heat-kernel and spectral-dimension bounds by established resistance-form techniques, although the authors deliberately leave those derivations for later work.","The internal exponents constructed in the proofs can be chosen non-increasing in $\\beta$, as recorded in Remarks 3.4 and 4.2, so stronger coupling does not make the resistance grow faster in $n$ within the constructed family of exponents."],"supporting_citations":[{"why":"Supplies the strengthened polynomial lower bounds for point-to-box and box-to-box resistances used as Lemma 3.3, giving the lower bound $\\delta>0$.","marker":"[11]"},{"why":"Provides the chemical-distance exponent $\\delta<1$ and the coarse-graining, animal-counting, and cut-point techniques adapted here to resistances.","marker":"[3]"},{"why":"Supplies the variational characterization of effective resistance used to compare resistances across different scales in Proposition 3.2.","marker":"[7]"},{"why":"Gives the BK inequality used to certify near independence of bad events in the coarse-graining argument.","marker":"[23]"},{"why":"Contributes the second-moment and cut-point method used to control the second moment of point-to-point resistances in Proposition 5.1.","marker":"[14]"}],"fun_headline_variants":["Resistances and box resistances share a single exponent δ in critical percolation","Effective resistances grow polynomially with exponent δ(β) in 1D critical percolation","Same δ for point-to-box and box-to-box resistances in critical percolation","One exponent governs all effective resistances in critical long-range percolation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on earlier results giving a uniform high-probability polynomial lower bound on resistances and a separate bound showing the growth exponent is below 1; if either of those imported bounds failed for some value of $\\beta$, the new exponent would not be guaranteed to lie in $(0,1)$.","fun_headline_variants_meta":{"raw":{"variants":["Resistances and box resistances share a single exponent δ in critical percolation","Effective resistances grow polynomially with exponent δ(β) in 1D critical percolation","Same δ for point-to-box and box-to-box resistances in critical percolation","One exponent governs all effective resistances in critical long-range percolation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00088,"raw_usage":{"total_tokens":3858,"prompt_tokens":1051,"completion_tokens":2807,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":2715}},"tokens_in":667,"tokens_out":2807,"duration_ms":23176,"temperature":1.0,"reasoning_tokens":2715,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:04:48.008584+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed $\\beta$ and large $n=2^k$, compute $\\mathbb{E}[R(0,[-n,n]^c)]$ (or estimate it by simulation over many samples) and check whether the ratio $\\mathbb{E}[R(0,[-n,n]^c)]/n^{\\delta}$ stays bounded away from $0$ and $\\infty$ as $k\\to\\infty$, where $\\delta$ is the slope of $\\log\\mathbb{E}[R(0,[-n,n]^c)]$ against $\\log n$. If that slope fails to converge to a value in $(0,1)$, or if the ratio diverges along a subsequence, Theorem 1.1 is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the strengthened polynomial lower bounds for point-to-box and box-to-box resistances used as Lemma 3.3, giving the lower bound $\\delta>0$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the chemical-distance exponent $\\delta<1$ and the coarse-graining, animal-counting, and cut-point techniques adapted here to resistances."},{"cited_title":"Biskup, J","cited_arxiv_id":null,"evidence_quote":"Supplies the variational characterization of effective resistance used to compare resistances across different scales in Proposition 3.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the BK inequality used to certify near independence of bad events in the coarse-graining argument."}],"review_version":1}