Infinite-Range Weighted Random Connection Models

• Infinite-range weighted random connection models are stochastic systems where vertices from a Poisson process carry weights that determine connectivity via a non-compact spatial decay kernel.
• The models integrate mark-based interactions with spatial decay functions, resulting in power-law degree distributions, rich phase transitions, and critical percolation thresholds.
• These models offer insights into diverse applications in stochastic geometry, statistical physics, and complex networks by capturing scale-free and inhomogeneous interactions.

Infinite-range weighted random connection models (WRCMs) generalize classical continuum percolation by allowing each vertex of a spatial Poisson process to carry a mark or weight, with edge probabilities determined jointly by the marks and spatial separation. These models exhibit rich phase transition phenomena, heavy-tailed degree distributions, and intricate connectivity and random-walk properties, capturing scale-free, long-range, and inhomogeneous interactions. The interplay of mark-distribution, kernel shape, and spatial-decay parameters yields critical thresholds and universal behaviors relevant to stochastic geometry, statistical physics, complex networks, and stochastic homogenization.

1. Formal Definition and Model Structure

Let $\eta$ be a homogeneous Poisson point process of intensity $\lambda > 0$ on $\mathbb{R}^d$, and let $M$ denote a measurable mark space equipped with probability law $\nu$ or $\pi$. Each $x \in \eta$ is assigned a weight or mark $m_x \in M$ independently. Edge formation between two vertices $x = (x, m_x)$ and $y = (y, m_y)$ is determined via a symmetric, measurable kernel: $$\varphi\bigl(x, y; m_x, m_y\bigr) = g(m_x, m_y)\, h(|x - y|)$$ where $g: M \times M \to [0, \infty)$ encodes mark- (weight-) dependence, and $h: \mathbb{R}_+ \to [0, 1]$ regulates spatial decay. Typical assumptions are non-increasing $h$, non-decreasing $g$ in each argument, and integrability: $\int_{\mathbb{R}^d} h(|z|)\, dz < \infty$ No compact support is imposed on $h$, resulting in infinite-range edge formation.

Variants utilize more general profiles or kernels, such as

• $p(x, y) = \rho\left( g(s, t)\, |x - y|^d \right )$ for real-valued normalization $\rho$ and distinguishing the roles—weight, radius, birth time—of the mark as parameterized by $\gamma$ and spatial-decay exponent $\delta$ (Gracar et al., 2019).
• The min-reach condition restricts the maximal connection range for a given weight: $\varphi(r; a, b) = 0$ if $r > R(\min\{a, b\})$ for some function $R$ (Caicedo et al., 25 Dec 2025).

2. Degree Distributions and Power-law Exponents

The degree distribution in infinite-range WRCMs is governed jointly by the mark distribution and the kernel structure. For kernels of power-law type (e.g., $g(s, t) = s^\gamma t^\gamma$), the expected degree for a vertex of mark $s$ is asymptotically proportional to $s^{-\gamma}$ as $s \to 0$: $\Lambda(s) \approx C s^{-\gamma}$ From the mark distribution (e.g., uniform $U(0,1)$ or Pareto-type tails), the unconditional degree tail exhibits a power-law: $$P(D > k) \asymp k^{-(\tau-1)}, \quad \tau = 1 + \frac{1}{\gamma}$$ Thus, for $\gamma > 0$, infinite-variance degree distributions emerge, with strong inhomogeneity and potential scale-free effects (Gracar et al., 2019). The tail exponent $\tau$ shifts under kernel choice and mark distribution, directly impacting critical thresholds and the prevalence of hubs.

3. Phase Transitions and Critical Thresholds

Percolation theory in WRCMs centers around the existence and uniqueness of an infinite component. The critical control parameter, often embedded as a density $\beta$ or intensity $\lambda$ in the kernel, marks the transition: $$\theta(\beta) = P^{0}[0 \text{ in infinite component}]$$ where $\theta(\beta) > 0$ implies percolation. For infinite-range models ($h$ not compactly supported, e.g., $h(r) \sim r^{-\delta}$ for $\delta > 1$), the critical threshold $\beta_c$ (or $\lambda_c$) is finite iff

$\gamma < \frac{\delta}{\delta + 1}$

for typical kernels; above this curve, percolation occurs for arbitrarily small density (“robust” regime), otherwise a nontrivial threshold is present (“non-robust” regime) (Gracar et al., 2019, Gracar et al., 2020).

Phase transition sharpness is characterized by exponential decay of the finite cluster-size distribution in the subcritical regime and linear growth of the percolation probability above $\lambda_c$ (Caicedo et al., 25 Dec 2025). The analysis employs OSSS inequalities, Russo–Margulis derivatives, and finite-lattice approximations to establish these properties rigorously.

4. Connectivity, Uniqueness, and Cluster Properties

Irreducibility and deletion-stability are central to the uniqueness of infinite clusters. A stationary marked WRCM (Poisson with marks and noncompact kernel) admits at most one infinite cluster almost surely, provided:

• Deletion-stability: removing a vertex does not split an infinite cluster into two disjoint infinite components.
• Irreducibility: any pair of points has a positive probability of being joined via finite paths.

Multiple rigorous criteria—atom, minorization, monotone-kernel—ensure irreducibility (Chebunin et al., 26 Mar 2024). Differentiability and convexity properties of the cluster density

$$\kappa(t) = \frac{1}{t} \mathbb{E}\left[ \#\, \text{finite clusters per unit volume} \right ]$$

yield analytic control and underlie uniqueness proofs via Mecke formula and Russo-type arguments.

5. Recurrence and Transience of Random Walks

The random walk on the infinite cluster behaves analogously to discrete long-range percolation. Let $G^\beta$ denote the infinite cluster at parameter $\beta$, regarded as an electrical network with unit conductances. Then:

• For preferential-attachment and similar kernels, transience holds whenever $1 < \delta < 2$ or $\gamma > \delta/(\delta + 1)$; recurrence in $d = 2$ for sufficiently slow spatial decay ($\delta > 2$) and small $\gamma$ (Gracar et al., 2019).
• For kernels with spatially long-range decay $h(r) \sim r^{-\alpha}$ and $\alpha \in (d, 2d)$, random walk is transient for large intensity (or robust regime); for $\alpha \geq 2d$, random walk is recurrent in low dimensions (Sönmez et al., 2019).

Key proof techniques employ multiscale renormalization, coarse-graining onto lattice models, Nash-type criteria, and conductance tests, establishing a correspondence with classical results of Berger and with phase diagrams dictated by $(d, \alpha)$.

6. Crossing Statistics, Diffusion, and Stochastic Homogenization

Exponential tail bounds for the number of disjoint crossings in large boxes ($\Lambda_L \subset \mathbb{R}^d$) of the infinite cluster are derived for marked WRCMs above the percolation threshold. These bounds ensure nondegeneracy of the homogenized diffusion matrix $D$ for simple random walks, random resistor networks, and exclusion processes: $$D(\rho) = \kappa(\rho) \mathbb{I}_{d \times d}, \quad \kappa(\rho) > 0$$ provided $\rho > \lambda_c$ (Faggionato et al., 5 Jul 2025). Applications extend to Poisson–Boolean models and Mott variable-range hopping networks, underpinning conductivity properties and universality features such as Mott's law.

Techniques include Tanemura’s growth process, seedless renormalization, uniqueness criteria, and two-scale homogenization arguments, integrating spatial and mark-induced long-range effects.

7. Cumulant Bounds, Subgraph Counts, and Edge-length Statistics

Analysis of subgraph counts and power-weighted total edge length leverages cumulant method bounds under appropriate moment conditions on weights, kernel tails, and mark distributions. For Poisson (and $\alpha$-determinantal) input processes, results include moderate deviation principles (MDP), central-limit estimates (CI), and negative association (NACC) for count statistics: $$|\kappa_m(Z_n)| \leq (m!)^{1+\gamma} \Delta_n^{m-2}$$ With suitable variance lower bounds and integrability, cumulant decay tames long-range spatial correlations and controls convergence (Heerten et al., 2023). Extension to determinantal point processes requires cycle-partitioning and kernel-integral bounds.

• "Recurrence versus Transience for Weight-Dependent Random Connection Models" (Gracar et al., 2019)
• "Connecting the Random Connection Model" (Iyer, 2015)
• "The random walk on the random connection model" (Sönmez et al., 2019)
• "Crossings and diffusion in Poisson driven marked random connection models" (Faggionato et al., 5 Jul 2025)
• "Explosion in weighted Hyperbolic Random Graphs and Geometric Inhomogeneous Random Graphs" (Komjáthy et al., 2018)
• "On the uniqueness of the infinite cluster and the cluster density in the Poisson driven random connection model" (Chebunin et al., 26 Mar 2024)
• "Percolation phase transition in weight-dependent random connection models" (Gracar et al., 2020)
• "Sharpness of the percolation phase transition for weighted random connection models" (Caicedo et al., 25 Dec 2025)
• "Lace Expansion and Mean-Field Behavior for the Random Connection Model" (Heydenreich et al., 2019)
• "Cumulant method for weighted random connection models" (Heerten et al., 2023)