Freezing a randomly chosen fraction of particles in an equilibrium reference configuration ('random pinning') is the standard equilibrium probe of amorphous order in glass-forming liquids. Mean-field theories in the random first-order transition (RFOT) class predict that pinning raises both the dynamical (mode-coupling) transition temperature TdT_d and the ideal Kauzmann temperature TKT_K, with the two lines merging at a terminal higher-order singularity conjectured to have random-field Ising criticality. Purely kinetic models with trivial statics — the Fredrickson–Andersen facilitated spin model on the Bethe lattice — were found in 2015–2025 to develop the same pinned phase diagram, including the terminal A3 point, through exact Bethe-lattice solutions; that criticality is purely dynamical, since the static susceptibility stays finite. The open problem is the finite-dimensional fate and universality of these pinning-induced singularities: whether randomly pinned liquids in two and three dimensions possess a genuine equilibrium transition line ending at a random-field-Ising-like critical point or only an avoided crossover; what the universality class of the terminal point is; and whether randomly pinned kinetically constrained models in finite dimensions remain singular at all. These jointly decide whether thermodynamic and purely kinetic slow dynamics respond to the pinning probe in the same way — the central equilibrium test of whether the glass transition is thermodynamic or kinetic in origin.

Background

When a liquid is supercooled below its melting point, its relaxation time grows by many orders of magnitude before it falls out of equilibrium and becomes a glass. Whether this slowing down announces a genuine thermodynamic phase transition — an 'ideal' glass transition reachable in equilibrium — or is a purely dynamical, kinetic phenomenon with no thermodynamic singularity behind it is arguably the central open question of glass physics. In the thermodynamic picture, realized by the random first-order transition (RFOT) theory and by fully connected spin-glass models with p-spin interactions (a p-spin model is a mean-field spin system whose Hamiltonian couples p spins at a time and whose glassy dynamics is exactly solvable), the configurational entropy of metastable amorphous states vanishes at a Kauzmann temperature TKT_K, while a dynamical transition to an ergodicity-broken phase occurs earlier at a mode-coupling temperature TdT_d (mode-coupling theory, MCT, is the dynamical theory predicting this transition). In the schematic MCT equation of motion the continuous dynamical transition corresponds to an 'A2' fold singularity, and the point where the transition line terminates is an 'A3' cusp singularity at which relaxation becomes logarithmically slow and the relaxation time diverges exponentially. In the opposing, purely kinetic picture, slow dynamics is generated entirely by local dynamical rules.

The archetype of purely kinetic slow dynamics is the Fredrickson–Andersen (FA) model, the simplest kinetically constrained model (KCM): a lattice of two-state spins ('mobile' and 'blocked') in which a spin can flip only when at least kk of its neighbors are mobile. Its equilibrium statics are trivial — the spins are independent — so any glassy phenomenology comes from the facilitation kinetics alone. On the Bethe lattice (a random regular graph that is locally tree-like, on which cavity, i.e. recursive message-passing, methods give asymptotically exact solutions) the FA model nevertheless develops a finite-temperature ergodicity-breaking transition.

Random pinning is an equilibrium protocol designed to probe amorphous order without quenching: an equilibrium configuration is sampled, a randomly chosen fraction cc of particles or spins is frozen in that reference configuration, and the remaining system is equilibrated in the field of the pinned matrix. Because the reference state is an equilibrium one, the pinned system can be studied below the temperature at which the unpinned liquid would fall out of equilibrium; the decay of correlations with distance from a pinned region defines the point-to-set correlation length, a standard measure of the spatial extent of amorphous order.

For mean-field glass formers, Cammarota and Biroli computed the pinned phase diagram: both Td(c)T_d(c) and TK(c)T_K(c) rise with the pinned fraction and the two lines merge and terminate at a finite-cc terminal point. The replica field theory of overlap fluctuations — the overlap qq measures the similarity of two independent copies of the system and is the natural order parameter for amorphous order — predicts that this terminal point has random-field Ising model (RFIM) criticality, the pinned particles acting as a quenched random field conjugate to the overlap (Franz, Parisi, Ricci-Tersenghi and Rizzo 2011; Franz and Sellitto 2013; Biroli, Cammarota, Tarjus and Tarzia 2014, 2018).

Whether the same phenomenology appears for purely kinetic models was posed by Ikeda and Miyazaki in 2015 and answered affirmatively at the Bethe-lattice level: in the FA model with facilitation k=2k=2 on a Bethe lattice of connectivity z=4z=4, random pinning raises the transition temperature and the transition line terminates at a critical point whose correlation-length exponents match the inhomogeneous MCT predictions at an A3 singularity (consistent with the critical dynamical heterogeneities computed by Nandi, Biroli, Bouchaud, Miyazaki and Reichman 2014). The full dynamics, solved by Ikeda, Miyazaki and Biroli in 2016, display the A2/A3 MCT critical behavior, with logarithmic relaxation and exponentially divergent relaxation time at the terminal point, and with self-induced disorder and avalanches strengthening the RFIM analogy.

Later exact work sharpened the contrast between the two sides: Perrupato and Rizzo (2024) solved the FA Bethe-lattice statics and found that the spin-glass susceptibility stays finite at the critical point — no static criticality, unlike RFOT-like systems; Leuzzi and Rizzo (2024) showed that the A3 logarithmic slowing down follows from the replicated Gibbs free energy in thermodynamic models but that in the pinned FA model there is no connection between the MCT-like dynamics and the connected correlation functions; Perrupato and Rizzo (2025) derived closed asymptotic MCT equations for FA dynamics for generic facilitation and connectivity including random pinning, concluding that criticality in the statics is not a necessary condition for criticality in the dynamics. At mean-field level, thermodynamic and kinetic glass formers therefore respond to pinning in closely analogous ways, but on the kinetic side the analogy is dynamical rather than thermodynamic.

What is not known is the finite-dimensional fate of all these predictions. Simulations of pinned liquids disagree about whether the transition lines rise and terminate as RFOT predicts or whether the apparent singularities are avoided crossovers; experiments in two-dimensional colloids see at most smooth crossovers; no study of randomly pinned kinetically constrained models in finite dimensions exists; and the random-field-Ising assignment for the terminal point has never been confirmed by fluctuation measurements. The Biroli–Bouchaud RFOT review (2024) lists exactly these as open issues.

Problem Statement

Determine whether the random-pinning glass transition established at mean-field level survives in finite dimensions, and whether pinning-induced singularities are generic across glass formers whose slowing down is thermodynamic versus purely kinetic. Concretely: (i) establish the equilibrium phase diagram of randomly pinned glass-forming liquids in two and three dimensions in the temperature–pinned-fraction plane — whether the dynamical and thermodynamic transition lines rise with pinned fraction and terminate at a terminal critical point, or whether the apparent singularities are avoided crossovers — thereby resolving the standing conflict between simulations that find the RFOT-predicted structure (Ozawa et al. 2015) and those that do not (Chakrabarty et al. 2015); (ii) determine the critical behavior and universality class of the terminal point, in particular whether overlap fluctuations obey random-field-Ising criticality as the replica field theory predicts, via fluctuation observables such as overlap and pinning susceptibilities, correlation lengths, point-to-set lengths, and avalanche statistics; and (iii) establish the effect of random pinning on kinetically constrained models in finite dimensions — beyond the exactly solved Bethe lattice — to decide whether purely kinetic slow dynamics also develops pinning-induced singularities or whether they are rounded by loop corrections, as related hybrid transitions are known to be. These three components are related open facets identified in the cited sources (Cammarota–Biroli 2012; Ikeda–Miyazaki 2015; Ikeda–Miyazaki–Biroli 2016; Biroli et al. 2014, 2018; Biroli–Bouchaud 2024), bearing on whether thermodynamic and purely kinetic glass formers respond in the same way to the random-pinning probe in finite dimensions; the objective is to fix the finite-dimensional fate and universality of the mean-field pinning predictions on these facets, and resolution of any single facet is not conditional on resolving the others.

References

  1. G. H. Fredrickson and H. C. Andersen, 'Kinetic Ising Model of the Glass Transition', Physical Review Letters 53, 1244 (1984), DOI: 10.1103/PhysRevLett.53.1244, https://doi.org/10.1103/PhysRevLett.53.1244
  2. S. Franz, G. Parisi, F. Ricci-Tersenghi and T. Rizzo, 'Field theory of fluctuations in glasses', The European Physical Journal E 34, 102 (2011), DOI: 10.1140/epje/i2011-11102-0, https://doi.org/10.1140/epje/i2011-11102-0
  3. C. Cammarota and G. Biroli, 'Ideal glass transitions by random pinning', Proceedings of the National Academy of Sciences 109, 8850 (2012), DOI: 10.1073/pnas.1111582109, https://doi.org/10.1073/pnas.1111582109
  4. C. Cammarota and G. Biroli, 'Random pinning glass transition: Hallmarks, mean-field theory and renormalization group analysis', The Journal of Chemical Physics 138, 12A547 (2013), DOI: 10.1063/1.4790400, https://doi.org/10.1063/1.4790400
  5. C. Cammarota, 'A general approach to systems with randomly pinned particles: Unfolding and clarifying the Random Pinning Glass Transition', EPL 101, 56001 (2013), DOI: 10.1209/0295-5075/101/56001, https://doi.org/10.1209/0295-5075/101/56001
  6. S. Franz and M. Sellitto, 'Finite-size critical fluctuations in microscopic models of mode-coupling theory', Journal of Statistical Mechanics: Theory and Experiment (2013) P02025, DOI: 10.1088/1742-5468/2013/02/p02025, https://doi.org/10.1088/1742-5468/2013/02/p02025
  7. W. Kob and L. Berthier, 'Probing a Liquid to Glass Transition in Equilibrium', Physical Review Letters 110, 245702 (2013), DOI: 10.1103/PhysRevLett.110.245702, https://doi.org/10.1103/PhysRevLett.110.245702
  8. S. K. Nandi, G. Biroli, J.-P. Bouchaud, K. Miyazaki and D. R. Reichman, 'Critical Dynamical Heterogeneities Close to Continuous Second-Order Glass Transitions', Physical Review Letters 113, 245701 (2014), DOI: 10.1103/PhysRevLett.113.245701, https://doi.org/10.1103/PhysRevLett.113.245701
  9. G. Biroli, C. Cammarota, G. Tarjus and M. Tarzia, 'Random-Field-like Criticality in Glass-Forming Liquids', Physical Review Letters 112, 175701 (2014), DOI: 10.1103/PhysRevLett.112.175701, https://doi.org/10.1103/PhysRevLett.112.175701
  10. S. Gokhale, K. Hima Nagamanasa, R. Ganapathy and A. K. Sood, 'Growing dynamical facilitation on approaching the random pinning colloidal glass transition', Nature Communications 5, 4685 (2014), DOI: 10.1038/ncomms5685, https://doi.org/10.1038/ncomms5685
  11. H. Ikeda and K. Miyazaki, 'Fredrickson-Andersen model on Bethe lattice with random pinning', EPL 112, 16001 (2015), DOI: 10.1209/0295-5075/112/16001, https://doi.org/10.1209/0295-5075/112/16001 (arXiv:1505.02912)
  12. M. Ozawa, W. Kob, A. Ikeda and K. Miyazaki, 'Equilibrium phase diagram of a randomly pinned glass-former', Proceedings of the National Academy of Sciences 112, 6914 (2015), DOI: 10.1073/pnas.1500730112, https://doi.org/10.1073/pnas.1500730112
  13. S. Chakrabarty, S. Karmakar and C. Dasgupta, 'Dynamics of Glass Forming Liquids with Randomly Pinned Particles', Scientific Reports 5, 12577 (2015), DOI: 10.1038/srep12577, https://doi.org/10.1038/srep12577
  14. S. Chakrabarty, S. Karmakar and C. Dasgupta, 'Vanishing of configurational entropy may not imply an ideal glass transition in randomly pinned liquids', Proceedings of the National Academy of Sciences 112 (2015), DOI: 10.1073/pnas.1512745112, https://doi.org/10.1073/pnas.1512745112
  15. C. Cammarota and B. Seoane, 'First-principles computation of random-pinning glass transition, glass cooperative length scales, and numerical comparisons', Physical Review B 94, 180201 (2016), DOI: 10.1103/PhysRevB.94.180201, https://doi.org/10.1103/PhysRevB.94.180201
  16. H. Ikeda, K. Miyazaki and G. Biroli, 'The Fredrickson-Andersen model with random pinning on Bethe lattices and its MCT transitions', EPL 116, 56004 (2016), DOI: 10.1209/0295-5075/116/56004, https://doi.org/10.1209/0295-5075/116/56004
  17. G. Biroli, C. Cammarota, G. Tarjus and M. Tarzia, 'Random-field Ising-like effective theory of the glass transition. I. Mean-field models', Physical Review B 98, 174205 (2018), DOI: 10.1103/PhysRevB.98.174205, https://doi.org/10.1103/PhysRevB.98.174205
  18. G. Biroli, C. Cammarota, G. Tarjus and M. Tarzia, 'Random field Ising-like effective theory of the glass transition. II. Finite-dimensional models', Physical Review B 98, 174206 (2018), DOI: 10.1103/PhysRevB.98.174206, https://doi.org/10.1103/PhysRevB.98.174206
  19. I. Williams, F. Turci, J. E. Hallett, P. Crowther, C. Cammarota, G. Biroli and C. P. Royall, 'Experimental determination of configurational entropy in a two-dimensional liquid under random pinning', Journal of Physics: Condensed Matter 30, 094003 (2018), DOI: 10.1088/1361-648X/aaa869, https://doi.org/10.1088/1361-648X/aaa869
  20. T. Rizzo, 'Fate of the Hybrid Transition of Bootstrap Percolation in Physical Dimension', Physical Review Letters 122, 108301 (2019), DOI: 10.1103/PhysRevLett.122.108301, https://doi.org/10.1103/PhysRevLett.122.108301
  21. T. Rizzo and T. Voigtmann, 'Solvable Models of Supercooled Liquids in Three Dimensions', Physical Review Letters 124, 195501 (2020), DOI: 10.1103/PhysRevLett.124.195501, https://doi.org/10.1103/PhysRevLett.124.195501
  22. L. Leuzzi and T. Rizzo, 'Logarithmic critical slowing down in complex systems: From statics to dynamics', Physical Review B 109, 174211 (2024), DOI: 10.1103/PhysRevB.109.174211, https://doi.org/10.1103/PhysRevB.109.174211
  23. G. Perrupato and T. Rizzo, 'Thermodynamics of the Fredrickson-Andersen Model on the Bethe Lattice', Physical Review E 110, 044312 (2024), DOI: 10.1103/PhysRevE.110.044312, https://doi.org/10.1103/PhysRevE.110.044312
  24. G. Perrupato and T. Rizzo, 'Theory of kinetically-constrained-models dynamics', SciPost Physics 18, 020 (2025), DOI: 10.21468/SciPostPhys.18.1.020, https://doi.org/10.21468/SciPostPhys.18.1.020
  25. G. Biroli and J.-P. Bouchaud, 'The RFOT Theory of Glasses: Recent Progress and Open Issues', Comptes Rendus. Physique 24, 9 (2024), DOI: 10.5802/crphys.136, https://doi.org/10.5802/crphys.136