Random Degree Constrained Process (RDCP)

• RDCP is a generalized random graph process where edges are sequentially added under fixed local degree constraints, capturing network evolution and percolation behavior.
• The process exhibits a sharp phase transition with a critical threshold, leading to the sudden emergence of a giant component and exponential decay in subcritical regimes.
• Analytical techniques such as combinatorial switching, spectral analysis, and ODE martingales are key to understanding RDCP dynamics and its divergence from uniform random graph models.

The Random Degree Constrained Process (RDCP) is a generalized random graph process in which edges are added sequentially to a host graph, subject to local degree constraints. This model captures a broad range of important phenomena in probabilistic combinatorics and statistical physics, underpinning the dynamics of random graphs with prescribed degree restrictions and the emergence of global connectivity under locally enforced limitations. RDCP encompasses processes on finite graphs (including the random $d$-process), infinite lattices (as Constrained-Degree Percolation), and disordered or random environments.

1. Formal Definition and Process Variants

Given a finite or infinite host graph $G = (V, E)$, each vertex $v$ is assigned a degree constraint $d(v) \in \mathbb{N}$, which may be uniform (homogeneous) or sampled iid from a probability vector (inhomogeneous). Edges are associated with independent random variables (frequently Uniform$(0,1)$ or exponential clocks) indicating their candidate "opening" times. The sequential process proceeds as follows:

• At time $t$, an edge $e = (u,v)$ attempts to open only if both endpoints satisfy $\deg_{G_t}(u) < d(u)$ and $\deg_{G_t}(v) < d(v)$.
• Once open, an edge remains open.
• The collection of open edges at time $t$ forms the evolving RDCP subgraph $G_t$.
• The process runs until no further feasible edge remains.

For finite graphs with prescribed degree sequence $D_n = (d_1, ..., d_n)$, the process produces edge-addition sequences ending when all vertex degrees reach their respective constraints or all feasible pairs are exhausted (Molloy et al., 2022). On infinite lattices, the process is typically analyzed in the continuous-time limit and viewed through the lens of percolation theory (Hartarsky et al., 19 Sep 2025).

Notational overview (homogeneous constraint):

• On $\mathbb{Z}^d$, all vertices have degree cap $\kappa$; edges $e$ receive independent $U_e \sim \mathrm{Uniform}(0,1)$; $e$ opens at $U_e$ iff both endpoints have degree $<\kappa$ at time $U_e^-$ (Hartarsky et al., 19 Sep 2025, Lima et al., 2020).
• For inhomogeneous models, $d(v)$ iid from $(p_j)$, where $P(d(v)=j)=p_j$.

2. Phase Transitions, Critical Times, and Sharpness

A defining phenomenon in RDCP is the occurrence of a phase transition associated with the sudden emergence of a macroscopic ("giant") component. For (possibly infinite) host graphs, the critical threshold is given by

$$t_c = \inf \{ t : \mathbb{P}(\text{origin is in an infinite cluster at time } t) > 0 \}$$

or, for the parameter $p$ in the continuous-time parametrization, the percolation threshold $p_c = \sup \{ p : \theta(p) = 0\}$, where $\theta(p) = \mathbb{P}(0 \leftrightarrow_p \infty)$ (Hartarsky et al., 19 Sep 2025, Amaral et al., 2020, Santos et al., 2021).

Key phenomena:

• Sharp phase transition: RDCP exhibits exponential decay of the one-arm probability (cluster radius) in the entire subcritical regime $p < p_c$ (i.e., in the absence of infinite clusters, the connectivity probabilities decay exponentially fast with distance). This was rigorously proved for RDCP on $\mathbb{Z}^d$ using a Duminil-Copin–Raoufi–Tassion randomized algorithm framework, despite the absence of FKG or product measure structure (Hartarsky et al., 19 Sep 2025).
• Minimal constraint threshold: On $\mathbb{Z}^d$, for homogeneous constraint $k$, $k=3$ is the minimal value yielding a nontrivial percolation transition; $t_c^d(k) < \infty$ iff $k \geq 3$ (Lima et al., 2020, Amaral et al., 2020).
• Universality and critical exponents: Numerical and analytic results show that for all $k \geq 3$, the correlation length exponent $\nu$ equals that of standard Bernoulli percolation, indicating RDCP lies in the same universality class for large enough constraints (Amaral et al., 2020).

3. Comparison with Uniform Random Graph Models

The distribution of the final graph produced by the RDCP can differ dramatically from classical uniform random graph models constrained to the same degree sequence:

• For nearly regular degree sequences, the final graph of RDCP is conjectured to be contiguous with the uniform random $d$-regular graph; this is the content of Wormald's contiguity conjecture, open for exactly $d$-regular sequences (Molloy et al., 2022).
• For non-nearly regular cases (e.g., a significant fraction of vertices have degrees distinct from the bulk), RDCP is statistically far from the uniform model: the total variation distance tends to $1 - e^{-\Omega(n)}$ (Molloy et al., 2022).
• Discrepancies can be observed in graph statistics, such as the counts of edges among low-degree vertices, where the two models differ by $\Theta(n)$ with high probability (Molloy et al., 2022).

4. Analytical Techniques and Local Limits

A. Combinatorial Switching Method

The process' non-product nature complicates direct enumeration and probabilistic calculations. A key advance is the adaptation of the "switching" method—classically used for enumerating combinatorial structures—to the growing random process context of RDCP. This enables precise double-counting arguments and control on distributional discrepancies between RDCP and uniform models (Molloy et al., 2022).

B. Spectral and Branching Process Characterization

For high-degree or "almost regular" hosts, the local weak limit of the RDCP has been rigorously established to be a multi-type branching process (MTBP), with the limiting local structure given by a recursive tree process or Poisson-weighted infinite tree (PWIT) object (Ráth et al., 2024). The emergence of a giant component is characterized by the principal eigenvalue of an associated branching operator, where the phase transition occurs when this eigenvalue crosses $1$ (Ráth et al., 2024, Ráth et al., 15 Jan 2026). For the random $d$-process, as $d \to \infty$, the critical time approaches $\tfrac12$ (up to exponential corrections).

C. Martingale and Differential Equation Methods

The evolution of degree statistics (e.g., the minimum degree, fractions of certain degrees) is described by self-correcting martingales and solutions to coupled ODEs. Toward the end of the process, random fluctuations occur on scales determined by logarithmic powers of $n$, with precise limiting distributions given by independent exponentials at successive degree-lifting times (Hofstad, 2023).

5. Inhomogeneous Degree Constraints and Random Environments

Generalizations to random or spatially variable constraints have been analyzed, especially in percolation-theoretic settings:

• For a lattice with constraints $\kappa_v$ sampled iid from $\{0,1,2,3\}$, the existence of a phase transition is governed by the density $\rho_3$ of sites with the maximal allowed degree; a non-trivial critical threshold $\rho_3^* \in (0,1)$ exists such that percolation occurs if and only if $\rho_3 > \rho_3^*$ (Sanchis et al., 2020).
• The process admits decoupling inequalities: local events separated by large distances become nearly independent, with explicit exponential tail estimates on influence regions (Sanchis et al., 2020).
• Techniques extend to sharp threshold results, exponential decay proofs, and continuity with respect to model parameters (Santos et al., 2021, Sanchis et al., 2020).

6. Outstanding Problems, Implications, and Universality

RDCP represents a critical paradigm in the study of random graph processes with local constraints, characterized by rich structural transition phenomena and intricate long-range dependencies.

Open questions and ongoing directions include:

• Resolving Wormald’s contiguity conjecture for exact regularity in RDCP versus uniform models (Molloy et al., 2022).
• Developing purely analytic (e.g., differential equation) approaches for RDCP–uniform discrepancies (Molloy et al., 2022).
• Extending combinatorial and spectral methods to broader dependent, dynamical, or glassy models (Hartarsky et al., 19 Sep 2025).

In the almost 2-regular regime, the critical time for the emergence of the giant component agrees to leading order with the Molloy–Reed formula, highlighting a form of universality even when final-model distributions are non-contiguous (Ráth et al., 15 Jan 2026). The entire subcritical phase displays exponential decay of connectivity probabilities, confirming the sharp nature of the transition and aligning with the general universality class of independent percolation (Hartarsky et al., 19 Sep 2025, Santos et al., 2021, Amaral et al., 2020).

7. Summary Table of Key Features

Aspect	Main Result/Description	Reference
Critical time/threshold	Sharp phase transition, $t_c^d(k) \in (0,1)$ for $k \geq 3$; $t_c^d(k)$ nonincreasing in $k$	(Amaral et al., 2020)
Sharpness	Exponential decay of one-arm probability in subcritical regime	(Hartarsky et al., 19 Sep 2025)
Local weak limit	Multi-type branching process (MTBP) structure for high-degree hosts	(Ráth et al., 2024)
Uniform vs RDCP	Large total variation for non-nearly-regular sequences; discrepancy in "small-edge" statistics	(Molloy et al., 2022)
Random environment	Critical density $\rho_3^*$ for percolation with iid constraints; decoupling and sharp phase transition	(Sanchis et al., 2020, Santos et al., 2021)
Minimum degree evolution	Phase-separated, independent exponential hitting times; self-correcting martingale analysis	(Hofstad, 2023)

The Random Degree Constrained Process thus serves as a central object of study unifying random graph dynamics with local interactions, percolation transitions, and nontrivial limit laws. Robust analytical methods—ranging from combinatorial switchings and ODE martingales to spectral and branching process theories—are essential tools for its rigorous analysis.