Long-Range Percolation Overview

• Long-range percolation is a probabilistic model where connections between distant sites decay as a power law, bridging local and nonlocal interactions.
• The model employs non-perturbative renormalization group methods to derive critical exponents, scaling limits, and logarithmic corrections near the phase transition.
• Its framework informs applications in statistical physics and complex network theory, offering insights into universality classes and dimensional crossover phenomena.

Long-range percolation is a class of probabilistic models in which edges are allowed between distant sites of a host graph, with connection probabilities decaying as a function of distance. These models interpolate between local (short-range) and highly nonlocal (mean-field) behavior and capture rich phenomena relevant both in statistical physics and the theory of complex networks. The paper of long-range percolation reveals intricate dependencies between model parameters—such as dimension, decay rate, and connectivity—leading to a diverse phase diagram with sharp regimes, critical exponents, and scaling limits.

1. Mathematical Framework and Model Definition

Long-range percolation is commonly formulated on $\mathbb{Z}^d$ or hierarchical lattices, where each unordered pair of distinct vertices %%%%1%%%% is joined by an open edge independently with probability

$p(x, y) = 1 - \exp\left(-\beta J(x, y)\right)$

where %%%%2%%%% is the percolation parameter, and $J(x, y)$ is a kernel controlling the decay with distance. A widely-studied kernel is

$J(x, y) \propto \|x - y\|^{-d-\alpha}$

with $\alpha > 0$ fixing the tail exponent. Smaller $\alpha$ leads to slower decay—"more long-range"—while large $\alpha$ recovers short-range percolation. Connection rules for hierarchical lattices or inhomogeneous models may generalize this dependence, but the qualitative structure of interactions remains similar: longer edges are rarer yet can have macroscopic impact.

The central physical observable is the existence and properties of infinite connected clusters as a function of $\beta$, with the critical point $\beta_c$ defined by

$$\beta_c = \inf\{\beta : \mathbb{P}_\beta(|C(0)| = \infty) > 0\}.$$

Substantial attention focuses on the critical regime $\beta = \beta_c$, where the system exhibits scaling behavior and universal critical exponents.

2. Critical Regimes and Phase Transitions

Long-range percolation exhibits a rich array of regimes, controlled primarily by two parameters: the spatial dimension $d$ and decay exponent $\alpha$. The critical behavior is organized as follows:

Regime	Condition	Scaling/Exponents	Scaling Limit
Effectively long-range, HD	$d > 3\alpha$	Mean-field exponents	Super-α-stable (α < 2) or super-Brownian (α ≥ 2)
Effectively short-range, HD	$d > 6$, $\alpha > 2$	Mean-field exponents	Super-Brownian motion
Long-range, CD (critical dim)	$d=3\alpha < 6$	Logarithmic corrections	Superprocess (with correction)
LR low-dim (LD)	$d/3 < \alpha < \alpha_c(d)$, $d < 3\alpha$	Hyperscaling, non-mean-field	Conformal-like structures

Here, HD, CD, and LD denote high, critical, and low dimension respectively. Mean-field (MF) exponents such as the volume tail exponent $1/2$ and $\eta=0$ hold above the corresponding upper critical dimensions. The crossover between long-range and short-range regimes, at a model-dependent $\alpha_c(d)$, leads to a change in the two-point function exponent: $$2 - \eta = \begin{cases} \alpha & \text{if } \alpha < \alpha_c(d) \ 2 - \eta_{\text{SR}} & \text{if } \alpha \geq \alpha_c(d) \end{cases}$$ where $\eta_{\text{SR}}$ is the nearest-neighbor short-range exponent.

Critical exponents obey hyperscaling relations in low-dimension: $$\begin{aligned} &\eta = 2 - \alpha \qquad \gamma = (2-\eta)\nu \qquad \Delta = \nu d_f\ &\delta = \frac{d+\alpha}{d-\alpha} \qquad d_f = \frac{d+\alpha}{2} \end{aligned}$$ where $d_f$ is the fractal dimension of large clusters, and $\delta$ characterizes the cluster tail: $$\mathbb{P}_{\beta_c}(|K| \geq n) \asymp n^{-1/\delta}$$ Logarithmic corrections appear at the upper critical dimension $d_c = \min\{6,3\alpha\}$, causing deviations from pure power-law scaling in the critical volume, two-point function, and three-point function (Hutchcroft, 26 Aug 2025).

3. Renormalization Group and Analytical Methods

The rigorous analysis of long-range percolation, especially at and near criticality, leverages a non-perturbative real-space renormalization group (RG) framework (Hutchcroft, 26 Aug 2025). The core procedure is as follows:

• Introduce a finite cutoff $r$ (only allow bonds up to length $r$), inducing a subcritical finite-range model $P_{\beta_c,r}$.
• Derive infinite systems of ODEs for moments of cluster size and spatial statistics using Russo's formula and the mass-transport principle, e.g.:

$$\frac{d}{dr} \mathbb{E}_{\beta_c, r}|K| \sim \beta_c r^{-\alpha-1} \left(\mathbb{E}_{\beta_c, r}|K|\right)^2$$

• Solving these ODEs yields sharp asymptotics, e.g. $$\mathbb{E}_{\beta_c, r}|K| \sim \frac{\alpha}{\beta_c} r^\alpha$$ in high dimensions.
• Correlation inequalities (tree-graph, BK, "universal tightness") and diagrammatic expansions secure the control needed for scaling limits.
• The method is non-perturbative for $d > 3\alpha$ and also applies to spread-out models in $d > 6$ under two-point function estimates.

The RG flow identifies scaling functions and critical exponents, with higher-order (second-order) analysis necessary to extract logarithmic corrections at marginal dimensions.

4. Scaling Limits and Cluster Geometry

At criticality, the large-scale structure of clusters converges to universal stochastic processes, contingent on $\alpha$ and $d$:

• For $\alpha \geq 2$ and high dimension, the spatial displacement of a uniformly chosen cluster point, after rescaling by the radius of gyration $\xi_2(r)$, converges to Brownian motion killed at an exponential time ("super-Brownian excursion").
• For $\alpha < 2$ and $d > 3\alpha$, the spatial scaling limit is a super-α-stable process (an integrated superprocess driven by an $\alpha$-stable Lévy motion).
• At the crossover ($\alpha=2$), scaling functions acquire logarithmic corrections, e.g.,

$\xi_2(r) \sim r \sqrt{\log r}$

• The scaling limit of the rescaled cluster as a random measure converges to the canonical measure of the corresponding superprocess, in both cases (Hutchcroft, 26 Aug 2025, Hutchcroft, 26 Aug 2025).

The precise structure of multi-point connectivity (e.g., the $k$-point function) in the LR-LD regime displays Möbius covariance: $$\tau_{\beta_c}(x_1,\ldots,x_k) \asymp S(x_1,\ldots,x_k)^{-(d-\alpha)/2}$$ with $S(\cdot)$ a conformally covariant set function, suggesting the scaling limit in the LR-LD regime captures aspects of conformal invariance (Hutchcroft, 26 Aug 2025).

5. Upper Critical Dimension and Logarithmic Corrections

At the upper critical dimension ($d = 3\alpha < 6$), critical exponents continue to match their mean-field values, but observables such as the cluster volume tail and multi-point functions gain explicit logarithmic corrections: $$\begin{aligned} &\mathbb{P}_{\beta_c}(|K|\geq n) \sim C (\log n)^{1/4}/\sqrt{n}\ &\mathbb{P}_{\beta_c}(x\leftrightarrow y) \asymp \|x-y\|^{-d+\alpha}\ &\mathbb{P}_{\beta_c}(x\leftrightarrow y \leftrightarrow z) \asymp \sqrt{\frac{\|x-y\|^{-d+\alpha}\|y-z\|^{-d+\alpha}\|z-x\|^{-d+\alpha}}{\log(1+\min\{\|x-y\|,\|y-z\|,\|z-x\|\})}} \end{aligned}$$ This structure parallels findings in hierarchical percolation but differs from conjectures for nearest-neighbor models at $d=6$ (Hutchcroft, 26 Aug 2025).

The hydrodynamic condition, requiring that the maximal cluster in a finite box does not overwhelm the expected cluster mass, is necessary in this marginal regime to validate RG-derived scaling and applies for the critical dimension (Hutchcroft, 26 Aug 2025).

6. Dimensional Crossover and Universality

A core insight of long-range percolation theory is the existence of multiple universality classes determined by the interplay of $\alpha$ and $d$. The eight identified regimes correspond to combining LR/SR interaction range with high, low, or critical dimension, as summarized below (Hutchcroft, 4 Oct 2025):

LR/SR	Low-dim (LD)	Critical-dim (CD)	High-dim (HD)
LR	1	2	3
mSR	4	5	6
SR	7	8	9 (not listed in source but inferred as standard SR-HD)

Each regime is characterized by different scaling exponents and, when at the marginal lines, logarithmic corrections. The effective dimension $d_{\text{eff}} = \max \{d, 2d/\alpha\}$ determines where these transitions occur. As $\alpha \to \alpha_c(d)$ from below, long-range universality yields to short-range scaling, with smooth crossover in exponents except at critical points.

7. Implications and Research Directions

Long-range percolation models provide foundational insights into universality, scaling, and phase transitions in random media with nonlocal connectivity. The precise RG-based classification, critical exponents, and explicit scaling limits now established resolve major conjectures in probability and statistical physics.

Outstanding research directions include:

• Detailed characterization of the crossover regime $\alpha = \alpha_c(d)$, especially for $2 < d < 6$, where no general method computes $\alpha_c(d)$ (Hutchcroft, 26 Aug 2025).
• Extensions to percolation on non-Euclidean or inhomogeneous graphs, cluster robustness, and multifractal properties.
• Further exploration of conformal invariance in long-range and inhomogeneous models.
• Application of these scaling results to network science, particularly in understanding dimension-dependent phenomena in spatial or heavy-tailed networks, with relevance for epidemic modeling and communication systems.

The mathematical techniques refined in this program—including non-perturbative RG, differential inequalities, coarse-graining, and diagrammatic expansions—are broadly transferable and may drive similar advances in other models with long-range interactions and complex geometry.