{"id":"8a583fa1-83ba-4471-b059-00b5171b9b2f","arxiv_id":"2505.24283","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"At the critical line, local weak limits of Potts and random cluster measures on locally tree-like expander graphs are exactly mixtures of the free and wired tree Gibbs measures, and any mixture weight is realizable.","lead":"This mathematics paper proves that on expander graphs that look locally like a regular tree, the critical Potts model converges to a mixture of just two infinite-volume Gibbs states, the free and wired states. It also shows that by carefully modifying random regular graphs one can obtain any desired mixing weight between the two states.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.11(iii) does not yield the claimed contradiction: the sprinkling lower bound m(β)+δ/40 lies inside the [MMS12] concentration interval around the larger value m(β′), so Proposition 3.7 is not established as written.","rationale":"The reader's weakest_assumption was Lemma 4.10, the monotonicity inequalities Δ_{d−1,f} > Δ_{d−1,w} and Δ_{d+1,f} < Δ_{d+1,w}. I checked Appendix B.6 and found that argument coherent: F is strictly increasing on [0,1] because N(x) ≥ N(1) = (q−1)e^β(e^B−1) > 0, and the function G has the claimed monotonicity for t<1 and t>1. Thus I do not adopt the reader's concern as the main one. Instead, the load-bearing issue is internal to Section 3.2: Lemma 3.11(iii) is needed to prove Proposition 3.8, which proves the exponential deviation bound Proposition 3.7, which underpins both Theorem 1.4 and Theorem 1.5. The contradiction in the proof of Lemma 3.11(iii) is invalid because the lower bound obtained after sprinkling is below the upper endpoint of the concentration interval guaranteed by [MMS12] for the larger parameter β′. This is an internal-consistency problem, not a disagreement with consensus, and it appears potentially repairable by a sharper sprinkling choice, so the appropriate verdict remains CONDITIONAL rather than REJECT. The paper has substantial independent scaffolding — the rank-2 approximation, Edwards–Sokal coupling, small subgraph conditioning, and the general architecture of the proof — but this specific step must be fixed or replaced before the main theorems can be regarded as fully proved.","tokens_in":47927,"tokens_out":20224,"duration_ms":238201,"concrete_test":"Keep the sprinkling parameter ξ explicit in the last paragraph of Lemma 3.11(iii). (1) Replace the bound |C_1| ≥ (m(β)+δ/40)n by the stronger |C_1| ≥ X_n(R) − ξn and check whether, with ξ ≤ δ/100, this exceeds m(β′)+δ/40; if it does not, the contradiction truly fails. (2) Verify the exact statement of [MMS12, Theorem 2.5] to confirm whether it gives a two-sided interval around ±m(β′); if it gives only a one-sided upper tail, the attempted contradiction is even less justified. (3) If the gap is real, test Lemma 3.11(iii) numerically for representative (d, q, β) to see whether the asserted bound on E[X_n(R)] holds; a counterexample would refute Proposition 3.8 as stated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The classification Theorems 1.4 and 1.5 rest on Proposition 3.7, whose proof in Section 3.2 relies on Lemma 3.11(iii). That lemma aims to show E_{ψ^w_n}[X_n(R)] ≤ (m(β)+δ/10)|V_n| for some R. The proof by contradiction chooses β′>β with m(β′) m(β), and hence m(β)+δ/40 < m(β′)+δ/40. The event threshold lies strictly inside the concentration interval (m(β′)−δ/40, m(β′)+δ/40) that [MMS12] guarantees with probability 1−o(1), so no contradiction follows. A strengthened bound |C_1| ≥ (m(β)+δ/20−ξ)|V_n| with ξ < δ/100 would repair the argument, but the proof as written only records the weaker m(β)+δ/40 bound. Therefore the upper-bound half of Lemma 3.11, and consequently Proposition 3.8 and the exponential deviation estimate, is currently unsupported.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the q-state ferromagnetic Potts model with external field and the associated random cluster model on expander graph sequences converging locally to the d-regular tree T_d. Its first main result (Theorems 1.4 and 1.5) claims that on the critical line every subsequential local weak limit is supported on a single mixture of the free and wired tree Gibbs measures, for all integers d,q at least 3 and for all real q>2 in the random cluster case. Its second main result (Theorem 1.6) claims that for every mixture weight alpha in [0,1] there exist locally T_d-like expander graphs whose critical Potts measures converge in probability to alpha times the free measure plus (1-alpha) times the wired measure. The proof strategy combines the Edwards-Sokal coupling, a rank-2 reduction to a zero-field Ising model, exponential deviation estimates on expanders, small-subgraph-conditioning asymptotics for random d-regular graphs, and a graph-modification procedure. The paper is clearly organized and contains substantial appendix calculations, but two load-bearing points in the written proofs are not currently justified.","tokens_in":48283,"tokens_out":20561,"duration_ms":228475,"significance":"If the gaps described below are repaired, the paper would resolve a question left open by BDS23 and HJP23 and would provide the first construction of arbitrary phase-coexistence mixtures on locally tree-like expander graphs. The manuscript makes non-black-box use of external results such as CvdH25, BDS23, GvVY16, MMS12, and KLS20, and the candidate free and wired measures come from independent characterizations; there is no indication of circular reasoning. The appendix contains detailed moment and small-subgraph-conditioning calculations, and the paper is honest about which estimates are imported. Strengths include the explicit description of the limit set, the explicit graph construction for arbitrary mixture weights, and the substantial technical apparatus. No machine-checked proofs or reproducible code are provided, and the numerical evidence in Section 4.1 is informal.","major_comments":[{"comment":"The claimed contradiction with [MMS12, Theorem 2.5] does not follow from the inequalities written. The proof chooses beta-prime > beta with m(beta-prime) < m(beta)+delta/100 and derives, via psi^{w-prime}, an event of probability at least 1/2 - o(1) on which ||/|V_n| is at least m(beta)+delta/40. But [MMS12] for beta-prime only gives ||/|V_n| in (m(beta-prime)-delta/40, m(beta-prime)+delta/40) with probability 1-o(1). Since m(beta-prime) > m(beta), the threshold m(beta)+delta/40 is strictly less than the upper endpoint m(beta-prime)+delta/40, so the event is compatible with the concentration interval. Thus no contradiction is obtained. Consequently Lemma 3.11(iii) is unproved, and with it the upper-bound half of Proposition 3.8 and the exponential deviation estimate Proposition 3.7, on which both Theorems 1.4 and 1.5 rest. A different argument, for example a direct upper-deviation estimate at beta itself, is needed.","section":"Section 3.2, Lemma 3.11(iii)"},{"comment":"The case split in the construction is internally inconsistent. The proof announces that it treats only the case star = w, which by definition means delta^w_k > delta^f_k for all large k. Immediately afterward it chooses K satisfying 1 < (1+delta^f_K)/(1+delta^w_K) < 1+n^{-1}, which requires delta^f_K > delta^w_K and is impossible when star = w. Moreover, for star = w both the factor (Delta_{d+1,f}/Delta_{d+1,w})^p and the cycle factor ((1+delta^f_K)/(1+delta^w_K))^x are below 1, so the construction cannot increase the prefactor to the target gamma as claimed. The argument as written can only work for star = f, in which case the modification should use Delta_{d-1} rather than Delta_{d+1}. The arbitrary-alpha conclusion of Theorem 1.6 is therefore not established by the present text.","section":"Section 4.3, proof of Theorem 1.6"}],"minor_comments":[{"comment":"In the display for phi^{w,B}_{G^*_n}[|C^*| approx psi |V_n|], the denominator should be Z^{RC,w,B}_{G^*_n}, not Z^{RC,w,B}_{G_n}; the notation in (28) is correct and should be used consistently.","section":"Equation (17) and Proposition 3.4 display"},{"comment":"The expression 'm(delta) - delta/40' in the displayed inequality should be 'm(beta) - delta/40'; this is clearly a typo but it appears in a load-bearing argument.","section":"Section 3.2, proof of Lemma 3.11(ii)"},{"comment":"The phrase 'if eta is (beta, eta, R)-good' should read '(beta, delta, R)-good', matching the definition of good configurations in Proposition 3.8.","section":"Section 3.2, proof of Proposition 3.7"},{"comment":"The inequality S^d < Q^d is asserted without proof in the chain S^d < Q^d < T/(C1+C2) < P^d < R^d; it follows from nu^f(1)2 for the random cluster model, and construct arbitrary mixture weights. If the proofs are correct, this settles the BDS23 conjecture in full generality. The mixture-construction theorem is a real technical achievement.\n\nThe problem is that the proof of Theorems 1.4 and 1.5 has a hole in Lemma 3.11(iii). The argument by contradiction picks beta' > beta with m(beta') < m(beta)+delta/100, sprinkles, and obtains an event ||/|V_n| >= m(beta)+delta/40 with probability at least 1/2 - o(1). The text claims this contradicts [MMS12, Theorem 2.5] for tau^{beta'}. It does not. Since m is increasing, m(beta') > m(beta), so the threshold m(beta)+delta/40 lies strictly inside the concentration interval (m(beta')-delta/40, m(beta')+delta/40) that [MMS12] guarantees. Concentration around m(beta') is perfectly consistent with the event having probability near 1/2. The upper-bound half of Lemma 3.11 is therefore unsupported, and Proposition 3.8, the exponential deviation estimate (Proposition 3.7), and consequently Proposition 3.2 and the main classification theorems rest on it. This is not a minor typo; the comparison is in the wrong direction.\n\nWhat the paper does well is real. The local rank-2 approximation and the FK-Ising sprinkling are likely reusable even if the gap is fixed. Theorem 1.6's modification procedure is a genuine construction, and Lemma 4.10 appears to be proved carefully in Appendix B.6. The unproved numerical claim in Section 4.1 (that delta_f^k > delta_w^k always) is explicitly flagged and is not needed for the proof of Theorem 1.6; it only motivates going beyond random d-regular graphs. That is a minor caveat, not a flaw.\n\nBottom line: this is a serious paper, and the missing piece may be repairable, but as written the main theorems are not established. It deserves a serious referee, and the authors should get a chance to fix Lemma 3.11(iii). I would not cite the main theorems as proved until that repair appears.\n\nRecommendation: send to peer review, but the referee report should ask for a complete and correct proof of the upper bound in Lemma 3.11, or a different argument for the upper tail.","headline":"Genuine new results, but the proof of the main classification theorems has a load-bearing gap in Lemma 3.11(iii).","tokens_in":48831,"tokens_out":5693,"would_cite":false,"duration_ms":62301,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B20","82B27"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any integers d,q≥3, the critical Potts measures on locally tree-like expander graphs have local weak limits that are exactly mixtures of the free and wired Potts Gibbs measures on the d-regular tree, and every mixture weight can be…","keywords":["Potts model","random cluster model","local weak convergence","phase coexistence","expander graphs","Benjamini-Schramm convergence","free and wired Gibbs measures","critical line"],"falsifier":"Evaluate the four quantities Δ_{d−1,f}, Δ_{d−1,w}, Δ_{d+1,f}, and Δ_{d+1,w} defined in Appendix B.6 at small parameters such as d=q=3 on the critical line; if Δ_{d−1,f} ≤ Δ_{d−1,w} or Δ_{d+1,f} ≥ Δ_{d+1,w}, the arbitrary-mixture-weight conclusion of Theorem 1.6 fails for the constructed graphs.","tokens_in":47702,"feed_emoji":"🎲","tokens_out":8834,"duration_ms":87391,"temperature":0.7,"pith_summary":"For integers d,q≥3, consider the q-state ferromagnetic Potts model with an external field on sparse graphs that locally resemble the d-regular tree and are uniformly expanding. This paper proves that along the critical line, every subsequential local weak limit of the Potts measures is a convex combination α $μ^{{f,β,B}}$ + (1−α) $μ^{{w,β,B}}$ of the free and wired Potts Gibbs measures on the infinite tree. It further proves the converse with full strength: for every α in [0,1] there is a sequence of such expander graphs whose Potts measures converge locally weakly in probability to exactly that mixture. The same mixture characterization is proved for random cluster measures for every real q>2. This settles, for all d,q≥3, the prediction that critical Potts measures on locally tree-like expanders exhibit strong coexistence of the disordered and ordered phases.","feed_headline":"Critical Potts limits on expanders are exactly free-wired mixtures","feed_subtitle":"Every local limit mixes the free and wired tree states, and any mixture weight is attainable.","key_machinery":"The argument runs through the Edwards-Sokal coupling, which identifies the Potts measure as the spin marginal of a coupled random-cluster measure. A rank-2 approximation reduces the random cluster partition function on G_n to the partition function of a zero-field Ising model on the same graph; on the critical line the external-field term cancels, and the candidate mixture weights ψ_f and ψ_w are identified with the two possible Ising magnetizations ±m(β*). Exponential deviation estimates for large FK-Ising percolation clusters on expanders, obtained by a sprinkling argument, then force the component containing the ghost vertex to have size close to either ψ_f|V_n| or ψ_w|V_n|, which pins the local weak limit to the family M or N. For Theorem 1.6, small-subgraph conditioning gives sharp asymptotics of the free and wired partial Potts partition functions on random d-regular graphs in terms of short-cycle counts, and a cavity-method calculation shows that deleting m edges and adding p vertices of degree d−1 or d+1 multiplies the ratio of free to wired partition functions by a power of Δ_{d⋆,f}/Δ_{d⋆,w}. The two strict inequalities Δ_{d−1,f} > Δ_{d−1,w} and Δ_{d+1,f} < Δ_{d+1,w} ensure the two operations push the ratio in opposite directions, so any target ratio can be attained.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.4: if G_n are uniform edge-expander graphs converging locally to the d-regular tree T_d, then for any (β,B) on the critical line R_c, any convergent subsequence of the Potts measures $μ_n^{{β,B}}$ converges locally weakly in probability to a single measure in the family M = {α $μ^{{f,β,B}}$ + (1−α) $μ^{{w,β,B}}$ : α ∈ [0,1]}. There are no limiting mixtures with more than two components, and no non-tree-like limit points. Theorem 1.6 sharpens this by showing the whole interval of mixture weights is realized: for every α ∈ [0,1] there exist uniform edge-expander graphs G_n with girth tending to infinity such that $μ_n^{{β,B}}$ converges locally weakly in probability to α $μ^{{f,β,B}}$ + (1−α) $μ^{{w,β,B}}$. The analogous statement for random cluster measures, Theorem 1.5, holds for every real q>2, not just integer q.","pith_inferences":["Editorial extension: the mechanism suggests a general principle that on expanders with locally tree-like geometry, whenever two Bethe-optimal fixed points coexist, the limiting Gibbs measure should be a mixture of the corresponding tree Gibbs measures; one could test this on other models with first-order transitions, such as the hard-core model at its uniqueness threshold.","Editorial extension: because arbitrary α is obtained by deleting and adding O(1) edges per graph, the local weak limit is not determined by the local graph structure alone; global expansion plus sparse microscopic modifications control the mixture weight, which may matter for algorithms that estimate partition functions from local statistics.","Editorial extension: the inequalities in Lemma 4.10 are the only numerical input deferred to the appendix; checking them by exact computation for small d and q would either confirm the tuning construction or reveal a new regime.","Proposal: simulate the critical Potts measure on the modified random d-regular graphs constructed in the paper and measure the fraction of vertices whose local statistics match the ordered phase; it should converge to the chosen α."],"forward_implications":["On every uniform edge-expander graph sequence with G_n converging locally to T_d, the critical Potts measures have no exotic local limits: the only possible limits are free-wired mixtures, and convergence occurs in probability rather than only along subsequences.","Strong phase coexistence is real for all d,q≥3: at criticality the disordered and ordered phases can coexist with any prescribed weight α and 1−α.","The mixture weight is determined by the ratio of the free to wired partial Potts partition functions, which on random d-regular graphs is expressed through short-cycle counts.","The same mixture description holds for random cluster measures with real cluster parameter q>2, so the result is not an artifact of integer spin states.","These results cover the full parameter range d,q≥3, extending earlier treatments that required either zero field with very large q or even d."],"supporting_citations":[{"why":"supplies the stochastic-ordering framework and the even-d case that this paper extends; its Lemma 5.4 is used to identify free and wired limits.","marker":"[BDS23]"},{"why":"establishes the zero-field, large-q case on expander graphs and the free/wired partition-function asymptotics that this paper generalizes.","marker":"[HJP23]"},{"why":"introduces the rank-2 approximation of random cluster partition functions that is adapted to the local setting.","marker":"[BBC23]"},{"why":"extends the rank-2 approximation to positive external field and defines the critical line R_c used throughout.","marker":"[CvdH25]"},{"why":"provides the Ising local weak limit and magnetization tightness that the exponential deviation argument refines.","marker":"[MMS12]"},{"why":"supplies the sprinkling and cluster-size estimates on high-girth expanders used for the FK-Ising deviation bounds.","marker":"[KLS20]"},{"why":"gives the small-subgraph conditioning and cavity computations for partial Potts partition functions on random d-regular graphs.","marker":"[GvVY16]"},{"why":"identifies the free and wired belief-propagation fixed points as the only candidates for the Bethe maximum.","marker":"[DMSS14]"}],"fun_headline_variants":["Potts limits on expanders are free-wired mixtures, all weights","Expander Potts limits: only free-wired mixtures, any weight","Critical Potts on tree-like expanders: exactly free-wired mixtures","Free-wired mixtures capture all critical Potts expander limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction of all mixture weights in Theorem 1.6 rests on the two strict inequalities Δ_{d−1,f} > Δ_{d−1,w} and Δ_{d+1,f} < Δ_{d+1,w}; if either failed for some d and q, deleting or adding edges would push the free-to-wired ratio in the wrong direction and the proof would not realize arbitrary α.","fun_headline_variants_meta":{"raw":{"variants":["Potts limits on expanders are free-wired mixtures, all weights","Expander Potts limits: only free-wired mixtures, any weight","Critical Potts on tree-like expanders: exactly free-wired mixtures","Free-wired mixtures capture all critical Potts expander limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000701,"raw_usage":{"total_tokens":3216,"prompt_tokens":1051,"completion_tokens":2165,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":2088}},"tokens_in":667,"tokens_out":2165,"duration_ms":15991,"temperature":1.0,"reasoning_tokens":2088,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:27:27.028638+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the four quantities Δ_{d−1,f}, Δ_{d−1,w}, Δ_{d+1,f}, and Δ_{d+1,w} defined in Appendix B.6 at small parameters such as d=q=3 on the critical line; if Δ_{d−1,f} ≤ Δ_{d−1,w} or Δ_{d+1,f} ≥ Δ_{d+1,w}, the arbitrary-mixture-weight conclusion of Theorem 1.6 fails for the constructed graphs.","supporting_citations":[],"review_version":1}