The conditional Curie–Weiss model of Opoku, Edusei and Ansah conditions the mean-field Ising model on fixed proportions s and r of spins being permanently committed to +1 and -1, the remaining proportion t = 1 - s - r being free; the free block undergoes a first-order transition at beta* = (1-s-r)^{-1}, with a second threshold beta** in the asymmetric region s != r. The authors pose as open the extension in which the committed proportions are random rather than deterministic: determine the effect of this disorder on the phase transitions in the associated quenched and annealed models; make precise the sense in which the extension realizes a random-field Ising-type model whose induced random field has nonvanishing mean, and characterize its critical behavior; and, as the graph-level generalization, determine phase-transition properties of Ising models on graphs decorated by a random three-colouring (one colour class fixed to +1, one to -1, one free).

Background

The Curie–Weiss (CW) model is the mean-field version of the Ising model: NN spins σi∈{−1,+1}\sigma_i\in\{-1,+1\} interact all-to-all, with Gibbs measure proportional to exp⁡(−βHN(σ))\exp(-\beta H_N(\sigma)) for HN(σ)=−12N∑i,j=1Nσiσj−h∑i=1NσiH_N(\sigma)=-\frac{1}{2N}\sum_{i,j=1}^{N}\sigma_i\sigma_j-h\sum_{i=1}^{N}\sigma_i, where β>0\beta>0 is the inverse temperature and hh an external field. Its specific magnetization m(β,h)=lim⁡N→∞N−1∑i=1N⟨σi⟩m(\beta,h)=\lim_{N\to\infty}N^{-1}\sum_{i=1}^{N}\langle\sigma_i\rangle is determined by the minimizers of an explicit variational free-energy function, and for β>1\beta>1 it exhibits a first-order phase transition: the limits of m(β,h)m(\beta,h) as h→0+h\to0^+ and h→0−h\to0^- disagree, so the magnetization jumps.

Opoku, Edusei and Ansah introduced a conditional Curie–Weiss model as a stylized model of opinion formation in a society polarized along two opinions, equivalently of binary choice with social interaction in the sense of Brock and Durlauf's discrete-choice framework. The vertex set {1,…,N}\{1,\dots,N\} is partitioned into three blocks of asymptotic proportions ss, rr and t=1−s−rt=1-s-r: spins in the first block are fixed to +1+1 (individuals permanently committed to opinion 1), spins in the second to −1-1, and spins in the third block are free. The model is the CW Gibbs measure conditioned on this event. The authors proved that the specific magnetization undergoes a first-order transition at β∗=(1−s−r)−1\beta^{*}=(1-s-r)^{-1}, and that in the region 0≤s,r<120\le s,r<\frac12 with s≠rs\ne r a second threshold β∗∗=1r−sarctanh⁡(r−s1−s−r)\beta^{**}=\frac{1}{r-s}\operatorname{arctanh}\big(\frac{r-s}{1-s-r}\big) separates three qualitatively different jump scenarios of the field-magnetization curve (a jump without phase change; an ordered-to-disordered change; a magnetization sign flip). Structurally, the conditional model is exactly a CW model on the free block with effective coupling 1−s−r1-s-r and effective field h+s−rh+s-r.

The paper closes by posing three linked open directions, stated as limitations of the deterministic-proportion analysis. First, study the model when the proportions are fixed according to some distribution rather than deterministically, and determine the effect of this disorder on the phase transitions in the associated quenched and annealed models. In disordered systems, quenched means the disorder is drawn once and held fixed while thermal averages are taken, the free energy then being averaged over the disorder, whereas annealed means spins and disorder are averaged together, i.e. the partition function itself is averaged; the two generally differ. Second, the extension should realize a random-field Ising-type model with a non-centered (nonvanishing-mean) random field, whose critical behavior should be characterized. Third, such models set the stage for Ising models on graphs decorated by a random three-colouring: each vertex is assigned one of three colours by a random process, all vertices of one colour class are fixed to +1+1, those of a second colour class to −1-1, and the third colour class is free — the free colour class being a union of site-percolation clusters (in site percolation, each vertex is independently assigned a state, here a colour, and connected same-colour vertices form the percolation clusters) — and one asks whether the Ising model on such a decorated graph has a phase transition.

The relevant prior theory concerns the random-field Curie–Weiss (RFCW) model, obtained from the CW model by replacing the deterministic field with independent random fields on sites. Bovier's monograph and a sequence of papers establish variational formulas for its free energy, large and moderate deviation principles, metastability asymptotics and propagation of chaos for general field distributions, including nonzero-mean ones. However, that theory concerns independent site-wise fields. The disorder contemplated here enters instead through the proportions of fixed spins, so that the free block's effective coupling 1−s−r1-s-r and effective field h+s−rh+s-r are jointly random — a structure not covered by the i.i.d. theory. A further active line, the block-spin and multi-group Curie–Weiss literature, generalizes the multi-block structure of the conditional model but keeps all block proportions deterministic; related work has instead placed disorder in the interaction graph (a random communication structure between two equal groups) or compared quenched and annealed disorder in dynamical binary-choice models. None of these addresses random committed proportions.

Problem Statement

Extend the phase-transition analysis of the conditional Curie-Weiss model of Opoku, Edusei and Ansah to the case in which the committed proportions are random, and to the graph-decorated setting, as follows.

Setup. Keep the model of the source: NN spins with Curie-Weiss interaction at inverse temperature β\beta and external field (bias) hh, conditioned on a three-block partition in which a proportion of spins is fixed to +1+1, a proportion to −1-1, and the rest are free. Let the proportions be random in each of the following two senses: (a) a macroscopic random environment, in which the pair (S,R)(S,R) with S,R≥0S,R\ge0, S+R≤1S+R\le1 is drawn from a specified distribution π\pi once per realization and the blocks carry those proportions; (b) a per-site random three-colouring, in which each vertex is independently colored ++ with probability ss, −- with probability rr and free with probability t=1−s−rt=1-s-r. Define the quenched model (disorder fixed during the thermal average, free energy then averaged over the disorder) and the annealed model (partition function averaged jointly over disorder and spins).

Required results. (1) Quenched and annealed thermodynamic limits: compute the N→∞N\to\infty limits of the quenched and annealed free energies (pressures) for both disorder implementations, with variational characterizations. (2) Effect of the disorder on the phase transition: determine the phase diagram, in particular the fate of the first-order transition at β∗=(1−s−r)−1\beta^{*}=(1-s-r)^{-1} and of the second threshold β∗∗=1r−sarctanh⁡(r−s1−s−r)\beta^{**}=\frac{1}{r-s}\operatorname{arctanh}(\frac{r-s}{1-s-r}) of the deterministic model (read at the coloring probabilities (s,r)(s,r) in implementation (b), and at the disorder's mean or typical proportions in implementation (a)), together with the classification of jump types in the field-magnetization curves, under each disorder implementation in both the quenched and the annealed sense; in particular, whether the disorder-averaged (quenched) free energy retains or loses the non-analyticity, and how the annealed phase boundary shifts. (3) Non-centered random-field interpretation: make precise the sense in which the random-proportion extension realizes a random-field Curie-Weiss/Ising-type model whose induced random field has nonvanishing mean, via the effective field h+s−rh+s-r and effective coupling 1−s−r1-s-r of the free block, and characterize its critical behavior, distinguishing it from the classical independent-site random-field Curie-Weiss theory. (4) Decorated-graph generalization: for the Ising model on a graph whose vertices carry the random three-colouring (one colour class fixed to +1+1, one to −1-1, the third (free) colour class dynamical), determine whether and where a phase transition occurs as β\beta varies. The source leaves the graph family open. The complete graph is the mandatory floor — the analysis must cover it, consistently with items (1)-(3) — but the complete-graph case does not by itself fully resolve item (4): full resolution requires either the general structure or at least one further family on which the question is well posed (for example dense random graph sequences, Zd\mathbb{Z}^d, or trees), and the analysis must state clearly which families it covers.

These items form one program with a single objective: to determine how disorder in the committed proportions changes the phase-transition behavior of the conditional Curie-Weiss model. Item (3) is the reformulation of items (1)-(2) in random-field language, and item (4) is the same question lifted from the complete graph to decorated graphs, where the free colour class is a union of genuine site-percolation clusters rather than a single block. Answers must not silently restrict the question: results for a single disorder distribution, a single point (s,r)(s,r), or a single graph family resolve the corresponding part only, unless accompanied by the general structure (for example a variational formula valid for arbitrary π\pi or arbitrary (s,r,t)(s,r,t)).

References

  1. Alex A. Opoku, Kwame Owusu Edusei, Richard Ansah. Phase Transition in Conditional Curie-Weiss Model. arXiv:1608.07363 (2016). DOI 10.48550/arXiv.1608.07363. https://arxiv.org/abs/1608.07363
  2. Alex Akwasi Opoku, Kwame Owusu Edusei, Richard Kwame Ansah. A Conditional Curie–Weiss Model for Stylized Multi-group Binary Choice with Social Interaction. Journal of Statistical Physics 171(1):106-126 (2018). DOI 10.1007/s10955-018-1988-y. https://doi.org/10.1007/s10955-018-1988-y
  3. Anton Bovier. Statistical Mechanics of Disordered Systems: A Mathematical Perspective. Cambridge University Press (2006). DOI 10.1017/cbo9780511616808, ISBN 9780521849913. https://doi.org/10.1017/cbo9780511616808
  4. W. A. Brock, S. N. Durlauf. Discrete Choice with Social Interactions. The Review of Economic Studies 68:235-260 (2001). DOI 10.1111/1467-937x.00168. https://doi.org/10.1111/1467-937x.00168
  5. Matthias Löwe, Raphael Meiners, Felipe Torres. Large deviations principle for Curie-Weiss models with random fields. arXiv:1112.5380 (2011). https://arxiv.org/abs/1112.5380
  6. Matthias Löwe, Raphael Meiners. Moderate deviations for random field Curie-Weiss models. arXiv:1206.0895 (2012). https://arxiv.org/abs/1206.0895
  7. A. Bianchi, A. Bovier, D. Ioffe. Sharp asymptotics for metastability in the random field Curie-Weiss model. arXiv:0806.4478 (2008). https://arxiv.org/abs/0806.4478
  8. Francesca Collet, Richard C. Kraaij. Path-space moderate deviation principles for the random field Curie-Weiss model. arXiv:1705.00988 (2017). https://arxiv.org/abs/1705.00988
  9. Zakhar Kabluchko, Matthias Löwe. Propagation of chaos in the random field Curie–Weiss model. Journal of Statistical Mechanics: Theory and Experiment 2024:113206 (2024). DOI 10.1088/1742-5468/ad8f2a. arXiv:2312.01866 (preprint lists the authors in the reverse order). https://arxiv.org/abs/2312.01866
  10. Yihan He, Han Liu, Jianqing Fan. Hidden Clique Inference in Random Ising Model I: the planted random field Curie-Weiss model. arXiv:2310.00667 (2023). https://arxiv.org/abs/2310.00667
  11. Holger Knöpfel, Matthias Löwe, Kristina Schubert, Arthur Sinulis. Fluctuation Results for General Block Spin Ising Models. Journal of Statistical Physics 178:1175-1200 (2020). DOI 10.1007/s10955-020-02489-0. https://doi.org/10.1007/s10955-020-02489-0
  12. M. Löwe, K. Schubert. Exact recovery in block spin Ising models at the critical line. Electronic Journal of Statistics 14 (2020). DOI 10.1214/20-ejs1703. https://doi.org/10.1214/20-ejs1703
  13. H. Knöpfel, M. Löwe, H. Sambale. Large deviations, a phase transition, and logarithmic Sobolev inequalities in the block spin Potts model. Electronic Communications in Probability 26 (2021). DOI 10.1214/21-ecp397. https://doi.org/10.1214/21-ecp397
  14. Matthias Löwe, Kristina Schubert, Franck Vermet. Multi-group binary choice with social interaction and a random communication structure—A random graph approach. Physica A: Statistical Mechanics and its Applications 556:124735 (2020). DOI 10.1016/j.physa.2020.124735. https://doi.org/10.1016/j.physa.2020.124735
  15. W. Kirsch, G. Toth. Limit Theorems for Multi-group Curie–Weiss Models via the Method of Moments. Mathematical Physics, Analysis and Geometry 25 (2022). DOI 10.1007/s11040-022-09433-6. https://doi.org/10.1007/s11040-022-09433-6
  16. Michael Fleermann, Werner Kirsch, Gabor Toth. Local Central Limit Theorem for Multi-group Curie–Weiss Models. Journal of Theoretical Probability 35:2009-2019 (2021). DOI 10.1007/s10959-021-01122-4. https://doi.org/10.1007/s10959-021-01122-4
  17. Alex Akwasi Opoku, Godwin Osabutey. Multipopulation Spin Models: A View from Large Deviations Theoretic Window. Journal of Mathematics 2018:1-13 (2018). DOI 10.1155/2018/9417547. https://doi.org/10.1155/2018/9417547
  18. G. Toth. Models of opinion dynamics with random parametrisation. Journal of Mathematical Physics 65 (2024). DOI 10.1063/5.0159643. https://doi.org/10.1063/5.0159643
  19. Arkadiusz Jędrzejewski, José F. F. Mendes. Unified Framework for Binary-Choice Dynamics: Analysis and Applications. arXiv:2607.06803 (2026). https://arxiv.org/abs/2607.06803