Constrained-Degree Percolation

• Constrained-Degree Percolation is a probabilistic model on graphs where only configurations satisfying strict local degree rules and global constraints are permitted.
• It exhibits nontrivial phase transitions, transitioning from subcritical regimes with finite clusters to supercritical regimes with a unique infinite cluster.
• Analytical techniques such as decision-tree algorithms, combinatorial pivotality transformations, and planar duality methods provide insights for network routing and statistical physics applications.

Constrained-degree percolation encompasses a class of probabilistic models on graphs and lattices in which not all possible configurations of occupied (open) sites or bonds are admissible, but instead only those that satisfy strict local compatibility rules or global degree constraints. Typically, this means that the open subgraph produced by the percolation process must have every vertex of degree below a prescribed threshold or must obey certain pattern exclusions at the local level. Interest in constrained-degree percolation is motivated by systems in statistical mechanics with hard local rules (such as dimer and vertex models), by complex network architectures with bounded resources (e.g., router capacity in communications), and by the mathematical challenge posed by non-independent, non-monotone dependencies.

1. Model Classes and Local Constraints

The central object is a random subgraph or configuration $\omega$ of a host graph $G=(V,E)$, where occupation variables (for edges or vertices) are not mutually independent: a configuration is admissible only if it satisfies prescribed constraints. Two principal modeling routes for constraints are prominent:

• Fixed Vertex Degree Constraint: Each edge $e$ is assigned a random “activation time” $U_e$ (often uniform on $[0,1]$). Edges “attempt” to open at their assigned $U_e$. An edge opens at $U_e$ if and only if both endpoints currently have strictly fewer than $k$ open incident edges. Mathematically, the process tracked at increasing time $t$ constructs $\omega_t$ by activating all such $e$ with $U_e\leq t$ where both $\deg_{U_e}(\mathrm{endpts}(e))<k$ (Lima et al., 2020, Hartarsky et al., 19 Sep 2025, Hartarsky et al., 2020, Arcanjo et al., 2024).
• Pattern-exclusion Model (Face or Local Rule Constraints): At the level of faces or neighborhoods (e.g., sets of four vertices around a face in $\mathbb{Z}^2$), only configurations from a restricted set are permitted. For the square lattice, classical examples include only permitting $(0000), (1111), (0011), (1100), (0110),$ and $(1001)$ configurations around each face (rather than all $16$ binary patterns) (Holroyd et al., 2015, Li, 2017). This imposes parity conditions on checkerboard faces, leading to strong local dependencies among edges or spins.

In both schemes, the configuration space $\Omega$ is a strict subset of product space $\{0,1\}^E$ or $\{0,1\}^V$, and the measure is supported only on configurations compatible with the constraints.

2. Phase Transition and Critical Phenomena

Constrained-degree percolation generally exhibits a nontrivial phase transition akin to classical percolation. There exists a critical threshold $t_c(k, d)$ (for time, bond occupation probability, or other control parameter) such that:

• For $t < t_c$, only finite clusters exist almost surely, with connectivity decaying exponentially in distance (“subcritical regime”).
• For $t > t_c$, there is almost surely a unique infinite cluster (“supercritical regime”), provided the system has sufficient symmetry and ergodicity (Lima et al., 2020, Sanchis et al., 2020, Hartarsky et al., 2020, Arcanjo et al., 2024, Hartarsky et al., 19 Sep 2025).

Quantitative bounds in high-dimensional lattices reveal $t_c^{(\kappa)}(d) \sim 1/(2d)$ as $d \to \infty$ and $\kappa$ large (Hartarsky et al., 2020). Monotonicity results confirm that $t_c$ is a non-increasing function of the constraint $k$ (Amaral et al., 2020). For two-dimensional square lattices ($\mathbb{Z}^2$) and $k=3$, the phase transition threshold lies strictly between $1/2$ (the value for unconstrained bond percolation) and $1$ (Lima et al., 2020).

In random environment variants, each vertex may have its constraint $\kappa_v$ sampled i.i.d. from a prescribed distribution, and a nontrivial transition persists as long as a sufficient fraction of sites have maximal constraint (Sanchis et al., 2020, Santos et al., 2021).

3. Absence of Infinite Clusters and Sharpness

For two-dimensional constraint models with planar duality (e.g., those with pattern exclusions), strong results on the absence of infinite clusters (and dual “contours”) at criticality are established. The main theorems assert that under (possibly weak) symmetry, translation invariance, and ergodicity—supplemented by a finite energy condition—one almost surely has

$$\mu(\{\omega\colon\text{there is an infinite cluster}\}) = 0,$$

with similar results for contours (interfaces) separating cluster types (Holroyd et al., 2015, Li, 2017). At criticality, the mean cluster size diverges ($\chi = \mathbb{E}|\xi| = \infty$), but all clusters are finite.

Recent work has demonstrated the sharpness of the phase transition in constrained-degree percolation on the hypercubic lattice: the one-arm probability decays exponentially in the entire subcritical phase,

$$\mathbb{P}_p(0 \leftrightarrow \partial \Lambda_n)\leq \exp(-c n)\qquad \text{for }p<p_c$$

for an appropriate constant $c>0$. The proof leverages the Duminil-Copin–Raoufi–Tassion randomized algorithm method combined with a sophisticated combinatorial “pivotality” transformation in the absence of FKG and finite energy properties (Hartarsky et al., 19 Sep 2025).

4. Uniqueness and Continuity

For fixed degree constraint $k$ and dimension $d\geq3$, it is established that the infinite cluster above the threshold is unique almost surely for $t<t^*<1$ (and typically up to $t=1$) (Lima et al., 2020, Arcanjo et al., 2024, Hartarsky et al., 2020). The standard Burton–Keane-type arguments require significant adaptation, because the model lacks finite energy and insertion tolerance properties. Proofs invoke translation invariance, combinatorial surgery in finite volumes, and novel “decreasing clusters” modifications.

Moreover, the time evolution process in CDP exhibits the property that, for any local event $A$, $t\mapsto \mathbb{P}(\omega_t^{-1}(A))$ is differentiable and hence continuous, extending earlier results for dynamics with finite-range dependence (Arcanjo et al., 2024). As a consequence, the percolation function $\theta(t)$ is continuous in the supercritical regime $(t_c, 1)$.

5. Extensions: Random Environments, Pattern Models, and Dynamics

Random environment CDP models assign a heterogeneous collection of constraints $\{\kappa_v\}$ with prescribed probabilities to the vertex set, leading to spatially varying local rules (Sanchis et al., 2020, Santos et al., 2021). Analysis of these models requires decoupling inequalities, continuity arguments for local events, and multi-scale (coarse-graining) renormalization schemes to overcome infinite-range dependencies.

Pattern-exclusion constrained models, such as dimer coverings or XOR Ising models, can be mapped to percolation models with contour representations, where “even-degree” constraints at dual lattice vertices are derived from allowed configurations (Holroyd et al., 2015, Li, 2017). The interplay between clusters and their bounding contours provides strong geometric control, especially via planar duality.

Related dynamic models (e.g., deterministic spins with kinetic constraints) exhibit transitions classified in the directed percolation universality class, confirming that constraints in the evolution can induce absorbing-state phase transitions even in deterministic or multiple-absorbing-state settings (Deger et al., 2022).

6. Methodologies and Analytical Challenges

Key analytical advances in the field include:

• Randomized Decision-Tree/Algorithmic Methods: The Duminil–Copin–Raoufi–Tassion algorithm relates crossing probabilities to revealment and influence for decision trees, with adaptations to the infinite-range, non-monotonic dependencies in CDP (Hartarsky et al., 19 Sep 2025).
• Combinatorial Pivotality Transformations: Inspired by Aizenman–Grimmett essential enhancements, the analysis tracks how modifying edge variables along “switching paths” compensates for the lack of independence and localizes pivotality for the purposes of sharp threshold analysis.
• Planar Duality and Contour Arguments: Dual lattices and mapping to interface structures with even-degree constraints permit combinatorial control of infinite cluster and contour counts in two-dimensional models (Holroyd et al., 2015).

Throughout, the absence of the FKG inequality and finite-energy property—standard in i.i.d. models—necessitates the development of new combinatorial and probabilistic tools for uniqueness, continuity, and sharpness proofs.

7. Applications and Connections

Constrained-degree percolation provides a natural framework for describing systems with hard local resource bounds: network routing with connection limits (router models), capacity-limited social or biological interaction networks, and statistical mechanics models with vertex or face local constraints (dimer, 1–2, vertex models).

Moreover, the mathematical techniques developed for CDP have broader applicability to other interacting particle systems, kinetically constrained models, and random constraint satisfaction problems where independence, monotonicity, or finite-energy do not hold. Through its concrete applications and its theoretical challenges, constrained-degree percolation continues to inform both probability theory and the understanding of phase transitions in statistical and network systems.