{"id":"c91875da-dc90-4604-b9f9-fcd8334900dd","arxiv_id":"2412.19281","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Phase transitions occur in the long-range RFIM in d=1 for 1<α<3/2 and in d=2 for 2<α≤3, including the critical α=3.","lead":"This paper proves that long-range random field Ising models in one and two dimensions show ordered phases when the interaction decays slowly enough and the random field is weak. It settles open questions about the critical interaction exponent in two dimensions and removes a restrictive condition in one dimension.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 2D proof depends on an unverified transfer of [3, Prop 3.30] to d=2; if the coarse-grained entropy exponent is not n/2^{2rℓ}, Theorem 1.2 loses its entropy control.","rationale":"The reader's weakest assumption — that the long-range contour system of [3] transfers to d=2, especially [3, Prop 2.10] and [3, Prop 3.30] — is precisely the point on which the 2D theorem depends. The manuscript states this transfer without proof, and the entropy bound is dimension-sensitive in a way that is not checked. I found no separate internal inconsistency that would invalidate the 1D argument or the α=3 energy estimates; the final 'ε large enough' line is a typo that does not affect the quantitative bound, which can be read as ε small. Because the transfer is plausible and likely correct, I do not move the verdict to REJECT; the appropriate action is to keep the CONDITIONAL verdict and require the authors to supply the missing d=2 verification, either by reproducing [3, Prop 3.30] with d=2 constants or by citing a version of [3] that explicitly covers all d. If the combinatorial test above reveals an entropy overflow, the verdict would need to move to REJECT, as Theorem 1.2 would lack its main machinery. The concrete test is designed to settle exactly this question.","tokens_in":40530,"tokens_out":20458,"duration_ms":197659,"concrete_test":"Obtain the proof of [3, Proposition 3.30] and re-derive it explicitly for d=2, tracking every dimension-dependent constant and every use of the factor 2^{drℓ}. Then run a combinatorial check at α=3: fix n, take ℓ such that 2^{rℓ} ≈ √n, and enumerate coarse-grained interiors B_ℓ(Int) over all contours with |γ|∈[n,2n) and M_{n,Q,ℓ}=2^{k+1}; compare the logarithm of the count to (3.6). If the count exceeds C(ℓ^{κ+1}n/2^{2rℓ}+M) by a factor growing in n, Proposition 3.5 is false, and Theorem 1.2 lacks its entropy control. Independently, test [3, Prop 2.10] on a two-component contour in d=2 at separation exactly M min{|V|}^{a/2} with α=3 to confirm the energy gain lower bound holds without modification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.2 is proved only by asserting that the long-range contour machinery of [3] applies to d=2 'directly' and that [3, Proposition 3.30] can be 'adapted verbatim' (Section 3, before Prop 3.2, and Prop 3.5). This is the central load-bearing assumption for the 2D result. In particular, the entropy bound (3.6) has the form exp{b3 ℓ^{κ+1} n/2^{2rℓ} + b3 M_{n,Q,ℓ}}, and the α=3 proof uses it with M_{n,Q,ℓ}=2^{k+1} in (3.31) to absorb the Gaussian fluctuation error. If the correct dimension-dependent counting for d=2 contains a larger factor — for example n^2/2^{2rℓ} arising from disconnected long-range interiors — then the term exp{-C6 n/(ε^2(ℓ+1)^2 2^{rℓ})} in (3.32) may no longer dominate the entropy contribution. The improved energy estimate Lemma 3.13 does not repair an entropy overflow. Since the paper supplies no derivation of the d=2 entropy bound, this is the least secure point of the central claim. The 1D proof is self-contained and appears internally consistent; the typo 'ε large enough' at the end of §3.2 is minor and does not by itself affect the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proves two phase-transition theorems for the long-range random field Ising model with interaction J_{xy}=|x-y|^{-\\alpha}: Theorem 1.1 covers d=1 with 1<\\alpha<3/2, and Theorem 1.2 covers d=2 with 2<\\alpha\\le3. Both theorems assert that for sufficiently small disorder strength \\varepsilon and sufficiently low temperature \\beta, the plus boundary-condition state has, with high probability, very small probability of seeing a minus spin at the origin, implying coexistence of the plus and minus limiting states. The one-dimensional proof is based on a new multiscale Peierls map with a balancing/erasing procedure, and it does not require a large nearest-neighbor interaction. The two-dimensional proof adapts the long-range contour framework of Affonso, Bissacot, and Maia [3], introducing interaction-dependent coarse-graining estimates (Propositions 3.6 and 3.9) and a new logarithmic energy gain in the critical case \\alpha=3 (Lemma 3.13).","tokens_in":40827,"tokens_out":13686,"duration_ms":127965,"significance":"If the proofs are correct, these results are natural and valuable completions of the low-dimensional picture: the one-dimensional theorem removes the large-J(1) restriction of Cassandro, Orlandi, and Picco, and the two-dimensional theorem settles the critical exponent \\alpha=3 as still ordered, in contrast to the one-dimensional critical value \\alpha=3/2 where uniqueness holds. The one-dimensional part is essentially self-contained and contains genuinely novel ingredients, especially the balancing procedure and the energy bound in Lemma 2.29 together with the coarse-grained entropy bound in Proposition 2.20. The two-dimensional part also contains a new and interesting idea for \\alpha=3, namely the level-wise interaction accounting through Q_\\ell in Lemmas 3.7-3.9 and Lemma 3.13. The main weakness is that the two-dimensional contour entropy and contour enumeration are imported from [3] by assertion rather than proved in d=2; since Theorem 1.2 depends on these estimates, the central claim carries a correctness risk that the authors need to address.","major_comments":[{"comment":"Theorem 1.2 depends on the assertion that the long-range contour machinery of [3] transfers to d=2 \"directly\" and that [3, Proposition 3.30] can be \"adapted verbatim\" to give Proposition 3.5. This is a load-bearing step, and dimension d enters in several places: the geometry of Lemma 3.12, the cube areas in (3.16) and (3.20), and the exponential factor 2^{2r\\ell} in the entropy bound (3.6). In particular, the first term \\ell^{\\kappa+1} n / 2^{2r\\ell} in (3.6) is a counting estimate for coarse-grained interiors; if the correct d=2 count were of order n^2/2^{2r\\ell}, the entropy contribution would not be controlled by the energy in (3.32). Please provide a full proof of Propositions 3.3 and 3.5 for d=2, or give a precise citation to a statement that covers d=2. The current \"directly\" and \"verbatim\" statements are not sufficient for the paper's main two-dimensional claim.","section":"Section 3, before Prop. 3.2 and Prop. 3.5"},{"comment":"In the \\alpha=3 proof, Proposition 3.5 is applied with M_{n,Q,\\ell}=2^{k+1} to control both |B_\\ell(Int_{\\ell,k})| and |B_{\\ell+1}(Int_{\\ell,k})|. The hypothesis (3.5) for level \\ell is immediate from (3.19) and the definition of Int_{\\ell,k}, but for level \\ell+1 it is not immediate: the manuscript does not show that Q_{\\ell+1}(Int) is bounded in terms of Q_\\ell(Int), and Lemma 3.8 only gives a bound on the sum over all levels. Please supply the missing argument, for instance via (3.20) together with a comparison of |\\partial C_{\\ell+1}| with |\\partial C_\\ell| plus the symmetric difference, or revise the counting in (3.31).","section":"Section 3.2, around (3.30)-(3.31)"}],"minor_comments":[{"comment":"The sentence \"the desired result follows by taking \\varepsilon large enough\" is inconsistent with the displayed bound: the sum \\sum_{Q\\ge2} C_8 C_{10} Q^{1-C_8/\\varepsilon^2}(\\ln Q)^2 converges only for \\varepsilon^2 < C_8/2, and the preceding inequalities in (3.32) also require \\varepsilon small. This should read \"taking \\varepsilon small enough.\"","section":"Section 3.2, final paragraph"},{"comment":"The notation \\hat{Int}(n,Q) in Proposition 3.5 does not display the dependence on \\ell and M_{n,Q,\\ell}, even though the bound (3.6) involves both quantities. Please make the dependence explicit in the statement and in the application in Section 3.2.","section":"Proposition 3.5 statement"},{"comment":"The constant c_1 is introduced as an arbitrary constant at least 10, but later the same symbol c_1 is used with different fixed values, for example in Proposition 2.18 and Corollary 2.21. This reuse of notation is likely to confuse readers and should be cleaned up.","section":"Section 2, beginning"},{"comment":"The claim that the argument extends to the q-Potts model is stated without proof. If it remains in the paper, it should be phrased as a remark indicating a plausible extension rather than a proved statement, or a proof sketch should be added.","section":"Remark 2.23"}],"recommendation":"major_revision","confidential_remarks":"The two-dimensional section is largely a specialization to d=2 of the authors' prior work [3], on which one of the authors is a coauthor. This is not improper, but it strengthens the need for the authors to state exactly which d=2 statements are new and to make the transfer checkable. The one-dimensional part appears self-contained and is the strongest part of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this is a serious paper that completes the 1D and 2D phase diagram for the long-range RFIM, and it deserves a careful referee. The 1D result is the cleanest part: the balancing Peierls map is genuinely new, and it removes both the J(1) restriction and the α-window of Cassandro–Orlandi–Picco, reaching all 1<α<3/2. The proof is self-contained. Lemma 2.29, the energy bound for balanced configurations, is the key new estimate and it looks right; the entropy bounds are also spelled out. I did not find a gap in that section.\n\nThe 2D section is where I want referee time. The contour machinery is imported from Affonso–Bissacot–Maia [3], built for d≥3, and the paper says, twice, that the proofs “apply to d=2 directly” or can be “adapted verbatim.” That may be true, but it is not demonstrated. Specifically, [3, Prop. 3.30] gives the coarse-grained entropy bound exp{b3 ℓ^{κ+1} n/2^{2rℓ} + b3 M}. The exponent n/2^{2rℓ} looks dimension-sensitive. If the correct d=2 counting gives, say, n^2/2^{2rℓ} for disconnected interiors, then the α=3 estimate in (3.32) no longer beats the entropy term, and Theorem 1.2 loses its control. The new energy estimate Lemma 3.13 does not repair an entropy overflow. So the onus is on the authors to either prove the d=2 version of Prop 3.30 or show why the dimension does not change the exponent. This is the one place where a central step is an assertion about another paper rather than a proof in this one.\n\nMinor: the last line of the α=3 proof says “taking ε large enough” when the argument needs ε small. A typo, but fix it. Remark 2.22 (alternative proof of the no-field Dyson phase transition) is plausible but only sketched.\n\nBottom line: if the d=2 entropy transfer is real, this is a complete and valuable contribution, and the 1D part alone is worthwhile. I would send it to a serious referee and ask for the d=2 transfer to be made explicit before publication.","headline":"A strong 1D Peierls argument plus a plausible 2D theorem that leans on an unproven transfer of contour entropy machinery from d≥3; deserves a serious referee.","tokens_in":41472,"tokens_out":3907,"would_cite":true,"duration_ms":35615,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B20","82B26","82B44","60K35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The long-range random field Ising model is proved to have a phase transition in d=1 for 1<α<3/2 and in d=2 for 2<α≤3, including the critical α=3.","keywords":["random field Ising model","long-range interactions","phase transition","Peierls argument","contour methods","coarse-graining","Gaussian disorder","low-dimensional statistical mechanics"],"falsifier":"A direct check of the two-dimensional critical case: for a large square of side $N$ in $d=2$ with kernel $|x-y|^{-3}$, compute the minimum over subsets $A$ with $|A|=|A^c|=N^2/2$ of $J(A,A^c)$. If any sequence of configurations gives $J(A,A^c)\\ll N\\log N$, the logarithmic energy lemma (Lemma 3.13) underlying the $\\alpha=3$ proof is false. An easier check is to verify whether the energy bound of the adapted $d\\ge 3$ contour proof, [3, Prop. 2.10], holds numerically for random configurations in $d=2$ at $\\alpha=3$.","tokens_in":40283,"feed_emoji":"🧲","tokens_out":12805,"duration_ms":111880,"temperature":0.7,"pith_summary":"This paper proves that the long-range random field Ising model, where spins interact with strength $|x-y|^{-\\alpha}$ and are subject to a weak i.i.d. Gaussian field, orders at low temperature in dimensions one and two. In one dimension the phase transition holds for every $1<\\alpha<3/2$, removing the earlier requirement that the nearest-neighbor coupling be large. In two dimensions it holds for every $2<\\alpha\\le 3$, including the borderline exponent $\\alpha=3$ where the usual energy estimates do not suffice. The results are stated as bounds on the magnetization at the origin under $+$ boundary conditions: for small disorder $\\varepsilon$ and large inverse temperature $\\beta$, $\\mu^+_{\\beta,\\varepsilon h}(\\sigma_0=-1)$ is arbitrarily small with high probability, so the $+$ and $-$ states are distinct. This fills the low-dimensional picture, complementing known uniqueness regimes where the random field destroys order.","feed_headline":"Phase transition proved for long-range random-field Ising in d=1,2","feed_subtitle":"Two theorems reach low-temperature order down to one dimension and through the critical α=3 in two.","key_machinery":"In one dimension the argument rests on a multiscale Peierls map: a balancing procedure that repeatedly erases isolated intervals (intervals densely populated by one sign and surrounded by the other) until the configuration is balanced, then flips the remaining minus spins in the smallest isolated interval containing the origin. The two load-bearing estimates are the energy gain $J(A_\\sigma,A_\\sigma^c)\\ge c_2|I_\\sigma|^\\theta$ with $\\theta>1/2$ (Proposition 2.13) and an entropy bound that controls the number of possible Peierls images at each scale (Proposition 2.20). In two dimensions the machinery is the long-range contour system of [3], where contours are elements of the finest $(M,a)$-partition of the set of incorrect points, together with a scale-by-scale coarse-graining. For $\\alpha<3$ a uniform boundary-count bound suffices; for $\\alpha=3$ the proof introduces a finer energy decomposition $Q_\\ell(A)$ along cube boundaries that sums to the full interaction $J(A)$, and a new lemma (Lemma 3.13) giving $J(C^+,C^-)\\ge b_8\\sqrt{m+1}\\ln(m+1)$ for any cube with $m$ minority spins.","core_discovery":"The paper's central claim is that low-temperature, weak-disorder phase transition occurs in the long-range random field Ising model in $d=1$ for $1<\\alpha<3/2$ (Theorem 1.1) and in $d=2$ for $2<\\alpha\\le 3$ (Theorem 1.2). Concretely, for any $c_1>0$ there are $\\varepsilon_1(\\alpha,c_1)>0$ and $\\beta_1(\\alpha,c_1)>0$ such that for $\\varepsilon<\\varepsilon_1$ and $\\beta>\\beta_1$, with probability at least $1-c_1$, the $+$ boundary-condition limit satisfies $\\mu^+_{\\beta,\\varepsilon h}(\\sigma_0=-1)<c_1$; the analogue in $d=2$ holds with $c_2>0$ in place of $c_1$. Since the same argument applies to the $-$ boundary condition by symmetry, the two extremal Gibbs measures are distinct almost surely, which is the standard meaning of phase transition. The borderline exponent $\\alpha=3$ in two dimensions is the nontrivial case: the interaction energy of a flipped region of side $N$ is only of order $N\\log N$, just above the random-field fluctuation $\\sqrt{|A|}\\approx N$, and the proof supplies a matching energy gain via a logarithmic isoperimetric bound.","pith_inferences":["The one-dimensional balancing procedure appears robust enough to handle power-law decaying external fields and $q$-state Potts spins, which would likely yield deterministic analogues of Theorem 1.1 in those settings.","The refined coarse-graining based on $Q_\\ell(A)$ may apply to other marginal long-range systems, such as the two-dimensional model with $\\alpha=3$ and a decaying deterministic field, where a similar $N\\log N$ energy window appears.","If the transfer of the $d\\ge 3$ energy and entropy bounds to $d=2$ at $\\alpha=3$ fails, a counterexample would likely be a contour whose interior has a large boundary-to-interaction ratio; testing that ratio for large contours directly would settle it.","The theorem does not quantify correlation lengths or critical exponents; a natural next step is to determine whether the $\\alpha=3$ case in $d=2$ exhibits logarithmic corrections to the correlation length."],"forward_implications":["In $d=1$, phase transition is established for every $1<\\alpha<3/2$ without any lower bound on the nearest-neighbor coupling, removing the restriction of the earlier 1D result.","In $d=2$, phase transition is established for all $2<\\alpha\\le 3$; in particular the critical exponent $\\alpha=3$ is ordered, settling the question raised in the literature.","For the purely deterministic long-range Ising model in one dimension, the same Peierls map gives long-range order for $1<\\alpha<2$, an alternative proof of the recent contour-based result (Remark 2.22).","The balancing argument is formulated for $\\pm1$ spins and extends to $q$-state Potts models for $q\\ge 3$, giving low-temperature long-range order (Remark 2.23).","Together with the known uniqueness thresholds for the short-range random-field model, these theorems delimit the low-dimensional phase diagram: order for $1<\\alpha<3/2$ in $d=1$ and $2<\\alpha\\le 3$ in $d=2$, with uniqueness outside, except that $\\alpha=3/2$ in $d=1$ remains unique."],"supporting_citations":[{"why":"Supplies the long-range contour system and the energy/entropy bounds in $d\\ge 3$ that the two-dimensional proof adapts to $d=2$, including the critical $\\alpha=3$.","marker":"[3]"},{"why":"Provides the coarse-graining procedure and the subgaussian random-field estimates that control the disorder in both the one- and two-dimensional proofs.","marker":"[22]"},{"why":"Gives the error-function subgaussian lemma (their Lemma 3.1) and the strategy of flipping the external field together with the spins, used throughout the Peierls arguments.","marker":"[18]"},{"why":"Introduces the long-range one-dimensional contour concept and the $1/r^2$ phase transition that inspire the contour system and the 1D balancing construction.","marker":"[23]"},{"why":"Is the previous one-dimensional long-range RFIM result, with extra restrictions on the nearest-neighbor coupling, which the present Theorem 1.1 improves.","marker":"[12]"},{"why":"Establishes the uniqueness thresholds for $d=1,2$ and for $\\alpha=3/2$ in $d=1$ that delineate the boundary of the new phase-transition regime.","marker":"[7]"},{"why":"Develops the multidimensional long-range contour system, the improved version of which is the basis of the contour argument used in two dimensions.","marker":"[2]"}],"fun_headline_variants":["Long-range order tames random fields in 1D and 2D Ising","Long-range Ising orders in 1D and 2D despite random fields","Phase transition in 1D and 2D, including critical alpha=3","Critical alpha=3 cracked: 1D,2D random-field Ising orders"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The 2D proof assumes without proof that the long-range contour energy and entropy bounds from the d≥3 construction carry over directly to d=2 for all 2<α≤3, including the critical α=3.","fun_headline_variants_meta":{"raw":{"variants":["Long-range order tames random fields in 1D and 2D Ising","Long-range Ising orders in 1D and 2D despite random fields","Phase transition in 1D and 2D, including critical alpha=3","Critical alpha=3 cracked: 1D,2D random-field Ising orders"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001813,"raw_usage":{"total_tokens":7149,"prompt_tokens":971,"completion_tokens":6178,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":6089}},"tokens_in":587,"tokens_out":6178,"duration_ms":42320,"temperature":1.0,"reasoning_tokens":6089,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:45:32.474552+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check of the two-dimensional critical case: for a large square of side $N$ in $d=2$ with kernel $|x-y|^{-3}$, compute the minimum over subsets $A$ with $|A|=|A^c|=N^2/2$ of $J(A,A^c)$. If any sequence of configurations gives $J(A,A^c)\\ll N\\log N$, the logarithmic energy lemma (Lemma 3.13) underlying the $\\alpha=3$ proof is false. An easier check is to verify whether the energy bound of the adapted $d\\ge 3$ contour proof, [3, Prop. 2.10], holds numerically for random configurations in $d=2$ at $\\alpha=3$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the coarse-graining procedure and the subgaussian random-field estimates that control the disorder in both the one- and two-dimensional proofs."},{"cited_title":"Ding and Z","cited_arxiv_id":null,"evidence_quote":"Gives the error-function subgaussian lemma (their Lemma 3.1) and the strategy of flipping the external field together with the spins, used throughout the Peierls arguments."},{"cited_title":"Fröhlich and T","cited_arxiv_id":null,"evidence_quote":"Introduces the long-range one-dimensional contour concept and the $1/r^2$ phase transition that inspire the contour system and the 1D balancing construction."},{"cited_title":"Cassandro, E","cited_arxiv_id":null,"evidence_quote":"Is the previous one-dimensional long-range RFIM result, with extra restrictions on the nearest-neighbor coupling, which the present Theorem 1.1 improves."},{"cited_title":"Aizenman and J","cited_arxiv_id":null,"evidence_quote":"Establishes the uniqueness thresholds for $d=1,2$ and for $\\alpha=3/2$ in $d=1$ that delineate the boundary of the new phase-transition regime."},{"cited_title":"Affonso, R","cited_arxiv_id":null,"evidence_quote":"Develops the multidimensional long-range contour system, the improved version of which is the basis of the contour argument used in two dimensions."}],"review_version":1}