Spread-Out Bernoulli Percolation

• Sufficiently spread-out Bernoulli percolation is a framework where connectivity extends beyond nearest neighbors to include long-range interactions across diverse graph structures.
• The approach highlights critical thresholds, phase transitions, and universal mean-field scaling that reconcile classical lattice models with complex, inhomogeneous environments.
• Methodological innovations such as bootstrap methods, renormalization group techniques, and spatial Markov processes underpin the precise analysis of connectivity and scaling phenomena.

Sufficiently spread-out Bernoulli percolation refers to percolation models in which connections are not limited to nearest neighbors but may extend over longer distances or along more complex structures, such as Boolean models, layered graphs, inhomogeneous environments, or nonunimodular and high-dimensional settings. This generalized framework not only subsumes classical percolation on lattices but also highlights universal properties, critical thresholds, phase transitions, and regularity phenomena that emerge when the underlying graph or interaction range is “spread out” or “diluted” in an appropriate sense.

1. Definitions, Models, and Universal Features

Sufficiently spread-out Bernoulli percolation encompasses a family of models in which the underlying connectivity graph or process allows for edges or interactions to occur at longer ranges, with the degree of “spread” typically controlled by a parameter (such as distance scale $r$, interaction radius $R$, or percolation probability $p$ assigned to more than just nearest neighbors). Key examples include:

• Bond and site percolation on spread-out versions of transitive graphs of polynomial growth, such as $G_r$ where two vertices in $G$ are joined in $G_r$ if their distance in $G$ is at most $r$ (Spanos et al., 26 Apr 2024).
• Boolean percolation models, in which balls of random (possibly unbounded) radius are placed at occupied sites, and connections are declared when balls overlap (Coletti et al., 2014, Coletti et al., 2014).
• Models on layered graphs or ladders (Cartesian products with $\mathbb{Z}$), with interactions along multiple axes or incorporating vertical and horizontal randomness (Lima et al., 2019, König et al., 2022).
• Long-range or spread-out contact processes, spread-out voter models, and high-dimensional percolation with large $L$ or $R$ (Ráth et al., 2017, Rath et al., 2019, Duminil-Copin et al., 4 Oct 2024, Engelenburg et al., 24 Oct 2025).
• Anisotropic and inhomogeneous bond/site percolation, where edge probabilities can depend on direction or local heterogeneity (Gomes et al., 2021).

A common feature of these models is the emergence of critical phenomena (phase transitions, exponents, decay of correlations) that interpolate between low-dimensional lattice-dependent, and high-dimensional “mean-field” or universal, behavior as the spread or degree increases.

2. Critical Thresholds and Scaling in Spread-Out Graphs

One of the central results for sufficiently spread-out models is the quantitative determination of the critical threshold $p_c$ for the emergence of an infinite cluster. For vertex-transitive graphs of superlinear polynomial growth, the critical probability for Bernoulli bond percolation on the spread-out graph $G_r$ satisfies

$$p_c(G_r) = \frac{1 + o(1)}{\deg(G_r)} \quad \text{as } r \to \infty,$$

where $\deg(G_r)$ is the degree of a typical vertex in $G_r$ (Spanos et al., 26 Apr 2024). This sharp asymptotic matches earlier results for $\mathbb{Z}^d$ (Penrose, Bollobás–Janson–Riordan) and confirms, in the spread-out limit, that the transition occurs when the average degree is of order $1/p$, as would be predicted from Erdős–Rényi random graphs and mean-field theory.

Analogous threshold behavior occurs for anisotropic bond percolation in high dimensions, where percolation occurs whenever the sum of edge parameters $\sum_{i=1}^d p_i$ exceeds a critical value, often $1/2$ or a related constant, modulo regularity conditions (Gomes et al., 2021). In Boolean percolation on doubling graphs, percolation is absent if the radii distribution has a finite moment of order equal to the Assouad dimension and the retention parameter $p$ is small enough (Coletti et al., 2014).

In layered and inhomogeneous percolation models, critical thresholds for percolation are shown to be continuous functions of inhomogeneity parameters, ruling out abrupt phase diagram discontinuities under mild geometric constraints (Lima et al., 2019).

3. Phase Transitions, Critical Exponents, and Mean-Field Behavior

Spread-out models in high dimension or with sufficiently large range invariably demonstrate mean-field scaling exponents—a hallmark of universality in percolation. For sufficiently spread-out percolation on $\mathbb{Z}^d$ with $d > 6$ and large $L$, the two-point connectivity function at criticality decays as

$\tau_{p_c}(0,x) \asymp |x|^{2-d},$

and the probability that the origin connects to distance $n$ decays as $n^{-2}$ (full space) and $n^{-3}$ (half space), in line with one-arm critical exponents $\rho = 2$ and $\rho = 3$ (Engelenburg et al., 24 Oct 2025). The triangle diagram is finite at criticality, another mean-field indicator (Duminil-Copin et al., 4 Oct 2024).

The methodology for establishing these exponents includes:

• Sharp two-point function estimates via Simon–Lieb inequalities, bootstrap methods, and random-walk convolution bounds (Duminil-Copin et al., 4 Oct 2024).
• Entropic techniques coupled with correlation length sharpness to derive up-to-constants one-arm exponents (Engelenburg et al., 24 Oct 2025).
• Translation of percolation cluster exploration processes into random walks with heavy-tailed increments, allowing the computation of hull volume and perimeter exponents in random maps: $n^{-1/4}$ for hull volume and $n^{-1/3}$ or $n^{-1/2}$ for the perimeter, governed by universal stable laws (Angel et al., 2013).

These critical exponents and thresholds are insensitive to underlying local details, depending only on coarse parameters such as average degree, volume growth, and the long-range interaction profile—the essence of universality in sufficiently spread-out regimes.

4. Structural Phenomena, Connectivity, and Component Uniqueness

Spread-out percolation models often reveal new phenomena in the geometry and uniqueness of infinite clusters:

• On planar graphs with sufficiently large minimal degree and proper embeddings, Bernoulli site percolation can produce infinitely many infinite clusters on the interval $(p_c^{site}, 1-p_c^{site})$, confirmed using embedded trees and exponential decay arguments (Li, 2023).
• In Bernoulli line and hyperplane percolation, removing entire lines or planes induces power-law decay of connection probabilities and allows for regimes where the number of infinite clusters is $0,1,\infty$ (Hilário et al., 2015, Aymone et al., 2020).
• On (non)unimodular quasi-transitive graphs, “heavy repulsion” holds: any two heavy clusters (those with infinite total invariant weight) can only touch at a “light” (finite weight) set, ruling out robust macroscopic overlap. This follows from weighted generalizations of classical isoperimetric and Cheeger inequalities, via the theory of measure-class-preserving equivalence relations (Bell et al., 12 Sep 2025).
• In nonamenable and nonunimodular settings, heavy percolation clusters are found to be indistinguishable under group-invariant properties, and the uniqueness threshold matches the connectivity decay threshold (Tang, 2018).
• In highly supercritical and sufficiently spread-out oriented percolation, infinite clusters are robust and deviations from typical behavior have exponentially small probability, explained by renormalization group arguments (Tzioufas, 2013).

Such results not only reveal the complexity of cluster interactions and boundaries, but also solidify the connection between geometry, group invariance, and percolative properties.

5. Analyticity, Differentiability, and Regularity Properties

In spread-out models, macroscopic observables such as the percolation density $\theta(p)$, susceptibility, and transport coefficients (diffusivity, conductivity) admit regularity far from criticality:

• The percolation density $\theta(p)$ is analytic as a function of $p$ in the supercritical regime for both nearest-neighbor and spread-out (long-range) models, demonstrable via combinatorial and complex-analytic techniques using multi-interface expansions (Georgakopoulos et al., 2018).
• The susceptibility is analytic in the subcritical interval on any transitive short- or long-range model, with each cluster-size probability $p_m(p)$ being entire (Georgakopoulos et al., 2018).
• The effective diffusivity and conductivity of the infinite cluster in supercritical Bernoulli percolation are shown to be $C^\infty$: infinitely differentiable in the parameter $p$ (Gu et al., 8 Jun 2025). This regularity arises from uniform control of finite-volume derivatives and renormalization via cluster-growth decompositions and hole separation on the geometry, within the stochastic homogenization framework.

These regularity results reinforce the spread-out scenario as one free from Griffiths-type singularities away from the transition and as exhibiting smooth dependence of macroscopic observables.

6. Methodological Innovations and Connections to Other Areas

A variety of powerful methodologies facilitate the analysis of sufficiently spread-out percolation:

• Spatial Markov and peeling process techniques for random planar maps, leading to universal threshold formulas (Angel et al., 2013).
• Bootstrap iteration and reversed Simon–Lieb inequalities as alternatives to lace expansion for mean-field bounds (Duminil-Copin et al., 4 Oct 2024).
• Entropic methods for differential inequalities, sharpened by correlation length control (Engelenburg et al., 24 Oct 2025).
• Renormalization group arguments, multi-scale analysis, and combinatorial coupling (especially in highly supercritical or inhomogeneous/infinite-dimensional models) (Tzioufas, 2013, Lima et al., 2019).
• Markov chain constructions to analyze monotonicity in layered/Cartesian product graphs (König et al., 2022).
• Techniques from measured group theory, such as invariant random partitions and measure-class-preserving equivalence relations, to handle nonunimodular and weighted connectivity properties (Bell et al., 12 Sep 2025).

Through these tools, the spread-out framework provides deep connections to geometric group theory, random graph theory, statistical physics (including the Ising model and random walks), and ergodic theory.

7. Applications and Future Directions

The theory of sufficiently spread-out Bernoulli percolation has broad applications and several open directions:

• Rigorous models of connectivity and resilience in complex networks, including networks with spatial, hierarchical, or random long-range links.
• Phase diagrams and robustness of phase transitions in disordered systems, including disordered media, epidemiological models, and information spreading.
• Extensions to dependent percolation and to random environments beyond classical independence assumptions.
• Further regularity analysis near and at criticality, especially the analytic structure of transport coefficients and the possibility of strong/weak universality in more general settings.
• Topological and geometric properties (uniqueness, ergodicity, repulsion) of infinite clusters in nontrivial group and metric space contexts.
• Connections to the metric and percolation dimension theory developed in parallel with spread-out models.

Advances in these directions are likely to continue revealing the profound connections between stochastic geometry, probability, and the universality class structure underlying percolation transitions across mathematical and physical systems.