Kinetically Constrained Models (KCMs)

Updated 24 October 2025

Kinetically Constrained Models are interacting spin and particle systems defined by local dynamical constraints that yield trivial statics but complex, emergent glassy dynamics.
They exhibit glassy relaxation, dynamical heterogeneity, and anomalous transport with relaxation times scaling nontrivially with vacancy density.
Applications span classical and quantum systems, enabling studies of phase transitions in trajectory ensembles, aging, and the breakdown of standard diffusion laws.

Kinetically Constrained Models (KCMs) are interacting particle or spin systems equipped with local dynamical constraints that restrict allowed transitions depending on the configuration of their neighborhood. Although the equilibrium measures are typically trivial (product Bernoulli measures for particle occupation or spin states), these constraints induce highly nontrivial dynamical behavior, including glassy relaxation, dynamical heterogeneity, anomalous transport, and coexistence of active and inactive space‐time phases. KCMs serve as canonical stochastic models for the paper of glassy dynamics in both classical and quantum settings, providing a platform for the rigorous mathematical analysis of slow, heterogeneous, and out-of-equilibrium relaxation phenomena.

1. Model Definition and Core Principles

KCMs are defined by specifying a Markov generator of Glauber-type evolution on a lattice or graph, subject to a constraint function $c_x(\eta)$ for each site $x$ . For spin models, the generic generator is

$(\mathcal{L}f)(\eta) = \sum_x c_x(\eta)\left[\pi_x(f) - f(\eta)\right],$

where $\eta$ encodes the configuration, $\pi_x$ is the local update (typically a flip with bias $q$ ), and $c_x(\eta)$ encodes whether the move at $x$ is allowed (Martinelli et al., 23 Oct 2025).

Constraints may be noncooperative (e.g., FA--1f: at least one neighbor is vacant) or cooperative (e.g., FA--2f, Spiral, North-East; update at $x$ only if multiple, possibly directionally chosen, neighbors are vacant). Conservative KCMs preserve particle number (e.g., Kob–Andersen model), while nonconservative models do not.

Despite their trivial equilibrium measures (statics), KCMs exhibit highly nontrivial dynamics. This denotes a core conceptual distinction: all glassy features are an emergent property of the dynamical rules ("dynamical facilitation"), not static frustration (Garrahan et al., 2010, Martinelli et al., 23 Oct 2025).

2. Ergodicity Breaking and Relaxation Timescales

The possibility of ergodicity breaking, and the scaling of relaxation times, is controlled by the geometry of the constraints and the density of facilitating sites $q$ . For noncooperative KCMs (FA--1f, East), the dynamics are ergodic for any $q > 0$ . For cooperative and critical models (FA--2f, Duarte, Spiral), there exists a critical density $q_c$ so that below $q_c$ the system can be dynamically blocked (Martinelli et al., 2012, Marêché et al., 2018, Hartarsky et al., 2019).

Relaxation time $\tau$ scales parametrically with $q$ :

FA--1f (noncooperative): $\tau \sim q^{-3}$ in one dimension, $\sim q^{-2}$ in $d\geq 2$ .
East (oriented): $\tau \sim \exp[\beta^2/(2\ln 2)]$ (hierarchical, super-Arrhenius).
FA--2f, critical models: $\tau \sim \exp(c/q)$ or even $\tau \sim e^{\Theta((\log q)^4/q^2)}$ (Duarte) as $q\to 0$ (Marêché et al., 2018).

These slow relaxations are linked to the necessity for cooperative rearrangements, leading to energy barriers not present in associated bootstrap percolation dynamics. Exact sharp asymptotics for infection (relaxation) times and spectral gaps have recently been established for both critical and supercritical rooted models (Marêché et al., 2018, Hartarsky et al., 2019, Hartarsky, 2021).

3. Dynamical Heterogeneity and Space-Time Phase Transitions

KCMs reproduce key experimental observations in glassy systems, most notably dynamical heterogeneity—a spatial and temporal coexistence of regions with disparate mobilities. Quantitative measures include the persistence field $p_i(t)$ and the four-point susceptibility $\chi_4(t) = S_4(k\to 0, t)$ , which captures the growing scale of dynamical correlations as $q\to 0$ or $T\to 0$ (Garrahan et al., 2010).

A deeper perspective comes from the thermodynamic formalism of trajectories: the probability of a dynamical activity $K$ (the total number of configuration changes in a time window) follows a large deviation principle, and the dynamical partition function

$Z_t(s) = \langle e^{-sK}\rangle \approx e^{t\psi(s)}$

displays a singularity at $s=0$ , signaling a first-order (discontinuous) transition between an active (fluid) and inactive (jammed) trajectory phase (Speck et al., 2010, Turci et al., 2010, Garrahan et al., 2010, Bañuls et al., 2019, Causer et al., 2020). This "space-time" phase transition is in ensembles of histories, not configurations, and is central to the understanding of dynamic facilitation and intermittent relaxation phenomena.

In driven settings or under a field (nonconservative driving, current), similar large deviation singularities arise for entropy production and particle current, with microscopic origins manifesting as transient shear-banding, blocking walls, and dynamical coexistence of flowing and blocked domains (Speck et al., 2010, Turci et al., 2010).

4. Transport, Diffusion, and Aging

KCMs exhibit anomalous transport, including decoupling between the self-diffusion constant $D$ and the structural relaxation time, breakdown of Stokes–Einstein relations, and aging phenomena. For a tracer in equilibrium KCM, under positive spectral gap (ergodicity), the tracer diffuses with a rigorously established diffusion coefficient $D$ , which:

Scales as $D\sim q^{k+1}$ for $k$ -facilitated models (e.g., $D\sim q^2$ in FA--1f).
Satisfies $D\sim \text{gap}$ (spectral gap) for the cooperative East model; in this case, $D$ falls off faster than any power of $q$ (Blondel, 2013).

Aging arises in intermediate time regimes: for instance, in the East model, the temporal "staircase" alternates between rapid rearrangements and prolonged stalling, leading to nontrivial two-time correlation functions (Hartarsky et al., 18 Dec 2024).

Hydrodynamic behavior of KCMs, when connected to reservoirs at differing densities, leads to nonlinear diffusion equations with a strongly density- and model-dependent diffusion coefficient. Correlations generated by out-of-equilibrium conditions result in a diffusion coefficient systematically smaller than predicted by a mean-field ("no-correlations") approximation (Teomy et al., 2016).

5. Out-of-Equilibrium Relaxation and Mixing

Rigorous results on KCMs starting from general (possibly far from equilibrium) initial conditions have recently advanced. Convergence to equilibrium is shown to be exponential in infinite volume and linear (precutoff) in finite domains with appropriate boundary conditions, for all models in the so-called "critical" universality class, provided facilitating density in the equilibrium measure is close to $1$ (Hartarsky et al., 2022, Hartarsky et al., 18 Dec 2024, Marêché, 2019). The principal methodology couples KCM dynamics to attractive contact processes and employs block renormalization, last-passage percolation, and Toom contour ideas.

For oriented models like East, the concept of the "distinguished zero" or front underpins explicit proofs of cutoff and central limit scaling for mixing times. For high- $q$ (vacancy density) regimes, even cooperative models can be tackled by robust coupling to auxiliary processes.

Despite this progress, general proofs of exponential or cutoff mixing for arbitrary $q > q_c$ remain elusive, particularly in the low facilitating density regime, with many fundamental open problems remaining for even the one-facilitated model (Martinelli et al., 23 Oct 2025, Hartarsky et al., 18 Dec 2024).

6. Quantum and Thermodynamic Extensions

Quantum KCMs inherit classical facilitation constraints but superimpose quantum coherent or dissipative dynamics. Key results include:

The emergence of a nested hierarchy of frozen states in large-coupling regimes, leading to slow relaxation explained by the spatial distance between activating regions and a tower of metastable plateaus in correlations. The relaxation times scale as powers of the coupling parameter, and dynamical heterogeneity is controlled by the spatial arrangement of facilitating configurations (Marić et al., 3 Oct 2025).
Purely dissipative dynamics with quantum-constrained jump operators establish long-lived quantum coherences, orders-of-magnitude slower relaxation for off-diagonal observables than for densities, and illustrate limitations of classical rate equations in capturing non-equilibrium quantum relaxation (demonstrated in Rydberg gases under EIT) (Olmos et al., 2014).

"Thermodynamic flavor" KCMs have been constructed by coarse-graining nontrivial statics into bond-type KCMs, blending landscape properties with kinetic facilitation. This allows direct analytic connection between the energy landscape, relaxation time, and fragility index, and yields analytic solutions for N-point correlators using Bethe ansatz (Ashwin, 2019).

Mappings between KCMs and mean-field spin glass or p-spin models (RFOT theory) have revealed formal equivalence at the level of the master equation, with dynamical slowing down in KCMs corresponding to metastable state proliferation in RFOT, though the static order parameter in KCMs is highly nonlocal (Foini et al., 2012). Tensor network approaches, especially variational matrix product states, now allow systematic paper of large deviation statistics, spectral properties, and spatial structure in KCM trajectory space (Bañuls et al., 2019, Causer et al., 2020).

7. Universality, Classification, and Open Problems

Recent advances have led to refined universality categories for critical two-dimensional KCMs, with up to seven distinct classes, distinguished by the scaling exponents of relaxation time and spectral gap as $q\to 0$ (Hartarsky, 2021). The identification and characterization of dominant relaxation mechanisms, especially the explicit construction of energy barrier–dominated pathways, completed a universality program for equilibrium critical KCMs.

Open problems remain pervasive. For one-facilitated models (FA–1f), whether arbitrary initial conditions (with a single infection) always converge exponentially fast for all $q>0$ remains unresolved. Cutoff phenomena and the precise finite-size scaling of mixing times, especially in higher dimensions and for general KCMs, are the subject of ongoing research (Hartarsky et al., 18 Dec 2024, Martinelli et al., 23 Oct 2025). Understanding aging, dynamic heterogeneity, and connections to bootstrap percolation in greater depth also represent active areas of investigation.

Summary Table: Asymptotic Behavior of Key Quantities in KCMs

Model/Class	Relaxation Time $\tau$ Scaling	Reference
FA–1f (1D)	$\sim q^{-3}$	(Garrahan et al., 2010)
FA–1f (tracer)	$D \sim q^{2}$	(Blondel, 2013)
East	$\tau \sim \exp[\beta^2/(2\ln2)]$ ; $D \sim$ gap	(1009.61131306.6500)
Supercritical rooted	$\tau \sim \exp[\Theta((\log q)^2)]$	(Marêché et al., 2018)
Duarte model	$\tau \sim \exp[\Theta((\log q)^4/q^2)]$	(Marêché et al., 2018)
Critical KCM	$\tau \sim\exp[\Theta(1)/q^{2\alpha}]$ (class-dependent)	(Hartarsky et al., 2019 Hartarsky, 2021)

Glossary

Dynamical facilitation: The central mechanism in KCMs where site updates require neighboring "facilitating" configurations, leading to emergent glassy behavior.
Ergodicity breaking: The loss of connectivity in the configuration space, typically when vacancy density falls below a threshold.
Bootstrap percolation: An associated deterministic process used to classify ergodic components and stationary measures in KCMs.
Space-time phase transition: A singular change between dynamical phases (active/inactive) characterized via large deviation functions in trajectory ensembles.
Metastable plateaus: Long-lived non-equilibrium states in quantum KCMs, whose duration is set by energy barriers and constraint structure.

Extensive references are available for further detail, notably (Speck et al., 2010, Turci et al., 2010, Garrahan et al., 2010, Martinelli et al., 2012, Marêché et al., 2018, Hartarsky et al., 2019, Hartarsky et al., 2022, Hartarsky et al., 18 Dec 2024, Martinelli et al., 23 Oct 2025).