{"id":"78f4fc7b-1944-41d7-9a65-bc0a7a9efc29","arxiv_id":"2607.16561","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact Glauber-dynamics phase diagrams and hysteresis-loop shapes are derived for spin-1 random-field Blume-Capel and Blume-Emery-Griffiths models on a complete graph.","lead":"This paper solves two three-state random-field magnetic models on a fully connected graph at zero temperature, for both equilibrium and non-equilibrium Glauber dynamics. It finds exact phase boundaries and shows hysteresis loop shapes are set by the zero-disorder dynamics while loop area shrinks with disorder.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"RFBEGM stability criterion uses the equilibrium Hessian rather than the Glauber-map Jacobian; for K≠1 the dynamics is not gradient, and requiring D=0 and 1-A-B=0 simultaneously is not the generic loss-of-stability condition.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the two-order-parameter stability criterion is asserted from the Hessian of the equilibrium free energy rather than derived from the Glauber map's Jacobian. My stress-test sharpens this: for K≠1 the mean-field dynamics is not a gradient flow, so the Hessian of f is not the correct stability operator, and the paper's simultaneous D=0 and 1-A-B=0 condition is not the generic condition for loss of local minimality. This affects the RFBEGM phase boundaries and the R_c=0 claims. However, the paper's simulations partially support several qualitative conclusions (e.g., K<0, Δ>0 has no ordered state), so the appropriate recommendation remains CONDITIONAL: the analytic RFBEGM boundaries should be either proven from the Jacobian or directly verified by simulations. Since the reader's verdict is already CONDITIONAL, no change is needed.","tokens_in":19379,"tokens_out":26974,"duration_ms":267340,"concrete_test":"Numerically integrate the deterministic Glauber mean-field equations m_{t+1}=F_m(m_t,q_t) and q_{t+1}=F_q(m_t,q_t) using Eqs. 18-19 for representative parameters (e.g., K=2, Δ=3 and K=-1, Δ=1), scanning R and starting from both m=0 and m=1. Record where the m=0 state first loses stability, where m≠0 appears, and compare with Eq. 53 and the R_c=0 predictions. Independently compute the eigenvalues of J-I at the fixed points; if a zero eigenvalue occurs at c=1/K (for K>1) or at a point where D=0 but 1-A-B>0, the paper's simultaneous-condition criterion is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The RFBEGM phase boundaries (Sec. VI, Eqs. 49-53) are derived from the Hessian of the equilibrium rate function f(m,q), not from the Jacobian of the actual Glauber mean-field map. This is load-bearing because the central claim of exact Glauber steady-state phase boundaries for RFBEGM rests on it. The concern is not merely that the equivalence is unproved; for K≠1 the mean-field map (F_m,F_q) from Eqs. 18-19 has Jacobian J=[[A+B, K(A-B)], [A-B, K(A+B)]], so the dynamics is governed by J-I. The vector field (F_m-m, F_q-q) is a gradient only if (K-1)(A-B)=0, which generically fails for nonzero m and K≠1. Moreover, the paper claims the transition occurs when both 1-A-B=0 and D=0 simultaneously. A fixed point ceases to be a local minimum when either the leading principal minor or the determinant of the Hessian vanishes; requiring both is overly restrictive and generically wrong. For example, at m=0 the eigenvalues of J-I are 2c-1 and 2Kc-1; for K>1 the q-direction can become unstable at c=1/K before the m-direction condition c=1 is reached. Thus the phase boundary Eq. 53 and the R_c=0 arguments in Sec. VI are not established by the Hessian calculation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies zero-temperature Glauber dynamics and equilibrium statistical mechanics of the random-field Blume-Capel (RFBCM) and Blume-Emery-Griffiths (RFBEGM) models on a fully connected graph. For each model it derives self-consistent mean-field equations for the magnetization m and quadrupole moment q, and an equilibrium rate function via large deviations. It claims that the Glauber steady-state and zero-temperature equilibrium fixed-point equations coincide, but that below a critical disorder strength R_c the steady state depends on the initial condition, while for R≥R_c the two descriptions coincide. Analytic phase boundaries are given for both models, including a criterion for the RFBEGM based on the Hessian of the equilibrium free energy. The paper also derives hysteresis-loop shapes and coercive fields in the presence of a uniform field, and reports numerical simulations on N=1000 complete graphs supporting several of the analytical predictions.","tokens_in":19689,"tokens_out":9849,"duration_ms":98029,"significance":"If the central claims are correct, the paper provides a substantial exact result: for two spin-1 random-field models, the non-equilibrium zero-temperature Glauber steady state is exactly solvable and agrees with the equilibrium state above a disorder-controlled critical value, with interesting exceptions such as R_c=0 and a disorder-induced crossover to RFIM-like behavior. The large-deviations derivation of the zero-temperature rate function and the exact analytic treatment of hysteresis loop shapes are valuable. The RFBCM part, where the stability criterion reduces to a scalar condition, is convincing and is corroborated by simulations. However, the two-variable stability criterion for RFBEGM is the load-bearing element for the paper's most novel claims, and it is asserted rather than derived from the actual Glauber map. The correctness of that criterion is doubtful for K≠1, so the RFBEGM phase diagrams and the R_c=0/crossover conclusions require substantial revision or re-derivation.","major_comments":[{"comment":"The stability criterion for RFBEGM is asserted as positive-definiteness of the Hessian L of the equilibrium free energy f(m,q), with the transition at 1−A−B=0 and D=0 simultaneously. This is not derived from the Jacobian of the Glauber mean-field map (18)-(19), which is J=[[A+B, K(A−B)], [A−B, K(A+B)]]. For K≠1, L is not I−J and the vector field is not a gradient unless (K−1)(A−B)=0. At m=0 the Jacobian has eigenvalues c and Kc, where c=√(2/(πR^2))e^{−z^2/(2R^2)}. For K>1 the q-direction eigenvalue crosses +1 at c=1/K, before the m-direction crossing at c=1. Therefore Eq. (53) (c=1) does not give the loss of stability of the m=0 fixed point for K>1; the correct linear-stability boundary is c=1/K for that eigenvalue. This directly affects the phase boundaries derived for K>0 in Sec. VI and the claimed R_c values in that region.","section":"Sec. VI, Eqs. (49)-(53)"},{"comment":"The argument that for K<0, z≠0, D≥0 requires c≥1 while 1−A−B≥0 requires c≤1, and therefore R_c=0, uses the Hessian positive-definiteness conditions rather than the eigenvalues of the actual Glauber map. For K<0 the q-direction eigenvalue is Kc<0; stability of the fixed point requires only |Kc|<1, which can hold for c>1 when |K|c<1. Thus the determinant condition of the equilibrium Hessian is not the correct linear-stability condition for the map. The conclusion R_c=0 for K<0, z<0 is therefore not established. The simulation evidence in Fig. 4(c) is at a single very small R and does not rule out a transition at finite R for the actual dynamics.","section":"Sec. VI, case K<0 and the R_c=0 statements"},{"comment":"The crossover from R_c=0 to R_c=√(2/π) is argued on the basis of the special z=0 line and Eq. (55). However, the boundary at which the system crosses from the z<0 regime to z=0 is not derived from the Jacobian of the two-variable map. The statement 'once q≤Δ/K the z becomes greater than 0... the system has a continuous order-disorder phase transition at R_c=√(2/π)' implicitly assumes that the m=0, z≥0 branch is stable according to the correct dynamics. Since the Hessian-based criterion is not the stability criterion for K≠1, the location and even existence of this crossover are not rigorously established by the paper's analysis.","section":"Sec. VI, crossover claim in case 2 (K<Δ<0)"}],"minor_comments":[{"comment":"There are several typographical errors: the title and text alternate between 'Grifitths' and 'Griffiths'; the abstract contains 'We also. consider' with a stray period; the Introduction says 'Gluaber dynamics'; Sec. IV has 'performming' and 'subsusbtituting'. These should be corrected.","section":"Throughout"},{"comment":"The notation 'ccosh' and 'cc=1' is unclear; presumably it means c·cosh and a condition involving c. Please define all symbols explicitly and avoid the ambiguous two-letter combination.","section":"Sec. VI, Eqs. (51)-(53)"},{"comment":"The caption for Fig. 1(b) appears to refer to both 'increasing Δ' and 'decreasing Δ' but does not clearly separate the two branches; consider splitting or clarifying the quasi-static protocol.","section":"Sec. V.B, Fig. 1 caption"},{"comment":"The derivation of hysteresis shapes at R=0 is clear, but the statement that for R≠0 'the shape of the hysteresis loop is retained until R_c' is only illustrated, not proven; a brief argument that the nullcline structure is preserved for small R would strengthen the claim.","section":"Sec. VII"}],"recommendation":"major_revision","confidential_remarks":"The most serious issue is the unproved and plausibly incorrect stability criterion for the two-variable RFBEGM analysis. If the authors can replace the Hessian criterion by a derivation from the Jacobian of the Glauber map, the RFBEGM phase diagram and the R_c=0/crossover results may change qualitatively, especially for K>1. I recommend that the editor require this re-derivation before acceptance. The RFBCM results and the hysteresis-shape analysis are likely sound and should be preserved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the RFBCM part is genuinely good; the RFBEGM part has a serious unproven step. The paper solves the T=0 Glauber steady state for the random-field Blume-Capel model on the complete graph, gets closed-form phase boundaries for the m=0 and m=1 initial states, derives hysteresis loop shapes with coercive fields, and supports it with simulations that match. That is a real, checkable contribution, and the hysteresis classification (rectangular, wasp-waisted, double) is a nice addition.\n\nThe soft spot is Sec. VI. For the two-order-parameter RFBEGM, the authors identify Glauber fixed points with minima of the equilibrium rate function f(m,q) and take the transition as the simultaneous vanishing of 1-A-B and det L. But that is not the stability criterion for the actual mean-field Glauber map F(m,q). The Jacobian of F is J = [[A+B, K(A-B)], [A-B, K(A+B)]], and the dynamics is a gradient flow only when K=1. For K>1, the q-direction goes unstable at c=1/K before the m-direction condition c=1 is reached; for K<0, the equilibrium Hessian is never positive definite while the map can still be stable. So the exact RFBEGM phase boundaries, including R_c=0 and the crossover to R_c=sqrt(2/pi), are not established by the calculation as written. The simulations may be right, but the analytic argument needs to either prove the equivalence for this specific dynamics or compute the boundary from J. This is load-bearing for the paper's central claim of exactness for RFBEGM.\n\nMinor issues: there is a missing factor of K in Eqs. 23 and 29 (Eq. 30 suggests it is a typo), and the equilibrium first-order line is obtained numerically, which is fine as a limitation.\n\nWho is this for: people working on mean-field random-field spin models and hysteresis. The RFBCM results are worth a serious referee even if the RFBEGM part needs major revision. I would send it to review, but the referee should push hard on Sec. VI. I would not cite the RFBEGM boundaries until the stability derivation is repaired.","headline":"Blume-Capel half is solid and worth your time; the RFBEGM stability analysis has a load-bearing gap because the Glauber map's Jacobian, not the equilibrium Hessian, controls the transition.","tokens_in":20192,"tokens_out":19911,"would_cite":false,"duration_ms":183769,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Zero-temperature Glauber dynamics in random-field spin-1 models is initial-state dependent below a critical disorder variance and matches equilibrium above it; phase boundaries follow from local stability of the equilibrium free energy.","keywords":["Glauber dynamics","random field","Blume-Capel model","Blume-Emery-Griffiths model","tricritical point","hysteresis","fixed point stability","complete graph"],"falsifier":"Simulate Glauber dynamics on a complete graph for the random-field Blume-Capel model at fixed R=0.2, starting from a state with small nonzero magnetization, and measure the quasi-static transition value of Δ; the claim fails if the system leaves the basin while g(m)>0 or if the transition occurs at a Δ different from the predicted Δ_c. For the Blume-Emery-Griffiths model with K<0 and z≠0, the claim R_c=0 is falsified if a nonzero steady-state magnetization persists for arbitrarily small R or if a transition appears at finite R away from the z=0 line.","tokens_in":19212,"feed_emoji":"🧲","tokens_out":6928,"duration_ms":71834,"temperature":0.7,"pith_summary":"The paper solves the random-field Blume-Capel and Blume-Emery-Griffiths models on a complete graph at zero temperature, both in equilibrium and under Glauber dynamics. It shows that the variance R of the Gaussian random field acts like temperature: above a critical value, the non-equilibrium steady state forgets its initial condition and coincides with the equilibrium state, while below it the steady state depends on where the dynamics started. The transition lines for the Glauber steady state are obtained exactly by requiring that a fixed point of the equilibrium free energy loses local stability, yielding closed-form equations such as Δ_c = sqrt(−R_c² log(πR_c²/2)). For repulsive biquadratic coupling, R_c can vanish, and in some regimes the model crosses over from R_c = 0 to the random-field Ising value sqrt(2/π). In a magnetic field, the same stability condition gives analytic hysteresis-loop shapes and coercive-field equations.","feed_headline":"Past a disorder threshold, Glauber dynamics matches equilibrium","feed_subtitle":"Below it, dynamics remembers its start; above it, disorder erases memory and phase boundaries become exactly computable.","key_machinery":"The engine of the argument is the identity between the fixed-point equations of the zero-temperature equilibrium rate function and the self-consistent steady-state equations of zero-temperature Glauber dynamics. Since each allowed Glauber move lowers energy, the dynamics is argued to remain in the basin of a local minimum of the equilibrium free energy; the transition under dynamics is therefore exactly where that fixed point changes from a local minimum to a maximum or saddle. For the Blume-Capel model this reduces to the second derivative g(m) = 1 − (1/√(2πR²))[exp(−(m−Δ)²/2R²) + exp(−(m+Δ)²/2R²)]; for the Blume-Emery-Griffiths model, the Hessian matrix of f(m,q) supplies two conditions, 1","core_discovery":"The central claim is that for spin-1 random-field models on a complete graph, the zero-temperature Glauber steady state is controlled by the local minima of the zero-temperature equilibrium free energy f(m,q). Whenever f has multiple minima (small R), the dynamics keeps the system in the basin of its starting minimum, so the steady state is initial-state dependent; when R is large enough that only one minimum remains, equilibrium and non-equilibrium coincide. The dynamical transition occurs exactly where a fixed point ceases to be a local minimum: for one order parameter, the condition is g(m) = 0, giving Δ_c = sqrt(−R_c² log(πR_c²/2)); for two order parameters, the Hessian conditions 1−A−B","pith_inferences":["A natural extension is to test whether the local-stability criterion survives at finite but small temperature; the paper notes the finite-temperature steady state is equilibrium-like, so initial-state memory would appear only as slow relaxation rather than in the steady state.","The R-as-temperature analogy suggests that response or avalanche statistics near R_c in these spin-1 models might inherit random-field Ising critical behavior, a prediction the paper does not develop.","Because the coercive-field equation is derived on a complete graph, comparing it with finite-dimensional simulations would show how sensitive the mechanism is to mean-field assumptions.","For the frustrated regime with negative K, where the dynamics is non-abelian, the coexistence of continuous and first-order segments inside hysteresis loops suggests avalanche statistics could differ qualitatively from the abelian random-field Ising model; the paper reports the shapes but does not analyze avalanche distributions here."],"forward_implications":["For R above the tricritical variance, the Glauber steady state and the equilibrium phase diagram coincide, so measurements of the steady state cannot distinguish equilibrium from athermal dynamics once disorder is strong enough.","Below that threshold, phase boundaries are path-dependent: increasing Δ from an m=0 start and decreasing Δ from an m=1 start give different transition points, so any protocol must specify the initial state and the direction of drive.","For repulsive biquadratic coupling, R_c can be zero, meaning the ordered state is unstable to arbitrarily weak disorder; increasing R can then induce a crossover to a transition at R_c = sqrt(2/π), giving the system a finite disorder threshold it did not initially have.","The analytic hysteresis-loop shapes (rectangular, wasp-waisted, parallelogram, hexagonal, double) are set by the R=0 dynamics; disorder shrinks the loop area and sets the coercive field through the same local-stability equation."],"fun_headline_variants":["Disorder threshold decides if spin dynamics forgets its start","Spin memory erased past critical disorder in random-field models","For random-field spins, memory persists until a disorder tipping point","Glauber dynamics matches equilibrium only past a critical field strength","Initial state matters in spin models until disorder exceeds a critical value"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole Glauber phase diagram rests on the assertion that zero-temperature energy-lowering dynamics keeps the system trapped in the basin of a local minimum of the equilibrium free energy, so a dynamical transition happens exactly when that minimum loses stability; the paper motivates this heuristically rather than deriving it from the Jacobian of the actual dynamical map.","fun_headline_variants_meta":{"raw":{"variants":["Disorder threshold decides if spin dynamics forgets its start","Spin memory erased past critical disorder in random-field models","For random-field spins, memory persists until a disorder tipping point","Glauber dynamics matches equilibrium only past a critical field strength","Initial state matters in spin models until disorder exceeds a critical value"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1360,"prompt_tokens":930,"completion_tokens":430,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":361}},"tokens_in":674,"tokens_out":430,"duration_ms":4665,"temperature":1.0,"reasoning_tokens":361,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T20:39:41.710549+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate Glauber dynamics on a complete graph for the random-field Blume-Capel model at fixed R=0.2, starting from a state with small nonzero magnetization, and measure the quasi-static transition value of Δ; the claim fails if the system leaves the basin while g(m)>0 or if the transition occurs at a Δ different from the predicted Δ_c. For the Blume-Emery-Griffiths model with K<0 and z≠0, the claim R_c=0 is falsified if a nonzero steady-state magnetization persists for arbitrarily small R or if a transition appears at finite R away from the z=0 line.","supporting_citations":[],"review_version":1}