Bernoulli Bond Percolation Overview

• Bernoulli bond percolation is a probabilistic model where each edge in a graph is independently declared open with probability p, forming connected open clusters.
• The model features a sharp phase transition at a critical probability p_c, distinguishing regimes with and without an infinite open cluster.
• Recent research expands the model to incorporate anisotropy, reinforcement, and inhomogeneity while exploring quantitative properties like chemical distance and connectivity decay.

Bernoulli bond percolation is the random subgraph obtained from a connected, locally finite graph $G=(V,E)$ by declaring each edge open with probability $p$ and closed with probability $1-p$, independently across edges. Its basic objects are the open clusters, that is, the connected components of the open-edge subgraph. On $\mathbb{Z}^d$ and on broader classes of infinite graphs, the model exhibits a phase transition at a critical parameter $p_c$: below $p_c$ infinite clusters do not occur, while above $p_c$ they occur with positive probability. Current research treats not only nearest-neighbor percolation on Euclidean lattices but also anisotropic, reinforced, and random-environment variants, together with quantitative questions about connectivity decay, chemical distance, analyticity, and the geometry of extremal finite clusters (Georgakopoulos et al., 2018, Gomes et al., 2021, Kobayashi, 18 Dec 2025).

1. Definitions, observables, and standard geometric structure

For Bernoulli bond percolation on a graph $G=(V,E)$, the percolation density $\theta(p)$ is the probability that a distinguished vertex belongs to an infinite cluster, the susceptibility is $\chi(p)=\mathbb{E}_p(|C(o)|)$, and the critical probability can be expressed as the threshold above which an infinite open cluster appears with positive probability (Georgakopoulos et al., 2018, Gomes et al., 2021). On $p$0, a central distinction is between ordinary connectivity and finite connectivity: in the latter, one conditions on the event that the connecting cluster is finite, which is the natural observable in the supercritical regime when a unique infinite cluster is present [(Campanino et al., 2011); (Lima et al., 2015)].

In supercritical Bernoulli bond percolation on $p$1, every edge is open with probability $p$2, and there exists almost surely a unique infinite open cluster $p$3 (Dembin, 2018). A principal metric observable is the chemical distance $p$4, defined as the length of the shortest open path in $p$5 joining $p$6 and $p$7. For $p$8, the scaled chemical distance satisfies

$p$9

where $1-p$0 is a deterministic norm on $1-p$1. Moreover, for any $1-p$2, there exists $1-p$3 such that for all $1-p$4 in $1-p$5,

$1-p$6

and the associated asymptotic shapes vary in Hausdorff distance with the same modulus (Dembin, 2018).

2. Critical thresholds and phase diagrams on different graph classes

On $1-p$7, rigorous upper bounds for bond-percolation critical probabilities can be derived through couplings with lower-dimensional or more tractable models. For non-oriented bond percolation, one such bound is $1-p$8, where $1-p$9 is the unique solution of an explicit equation involving $\mathbb{Z}^d$0; analogous bounds are available for oriented bond percolation, including recursive bounds across dimensions (Gomes et al., 2021). In high dimension, anisotropic non-oriented bond percolation admits a criterion close to the tree heuristic: if $\mathbb{Z}^d$1 and $\mathbb{Z}^d$2, then $\mathbb{Z}^d$3 for sufficiently large $\mathbb{Z}^d$4; without the regularity condition, $\mathbb{Z}^d$5 already implies percolation for every $\mathbb{Z}^d$6 (Gomes et al., 2021).

Beyond Euclidean lattices, critical thresholds can be expressed through graph geometry. For an infinite bounded-degree graph with contour constant $\mathbb{Z}^d$7 and a bi-infinite geodesic, one has

$\mathbb{Z}^d$8

while if the graph has no bi-infinite geodesic but $\mathbb{Z}^d$9 and wedge constant $p_c$0, then

$p_c$1

This criterion applies to amenable and non-amenable graphs and extends previous isoperimetric criteria (Alves et al., 2012).

On random planar lattices, Bernoulli bond percolation can have explicitly computable critical parameters. For Uniform Infinite Half-Planar $p_c$2-angulations, the critical threshold satisfies

$p_c$3

where $p_c$4 is a peeling parameter determined by the map class. The summary gives $p_c$5 for type-1 triangulations, $p_c$6 for type-2 triangulations, and $p_c$7 for quadrangulations, yielding $p_c$8, $p_c$9, and $p_c$0, respectively (Angel et al., 2013).

Phase diagrams become richer when multiple bond ranges are present. On oriented regular trees with short bonds of parameter $p_c$1 and long bonds of length $p_c$2 with parameter $p_c$3, the critical curve

$p_c$4

is continuous on $p_c$5 and strictly decreasing on $p_c$6; moreover,

$p_c$7

The model has a hybrid regime in which neither short nor long bonds percolate alone, but percolation occurs through paths using both types (Lima et al., 2017).

3. Connectivity functions, inverse correlation lengths, and finite-connectivity asymptotics

Two-point functions and their truncated counterparts encode the geometry of connections away from criticality. In the subcritical and supercritical extreme regimes on $p_c$8, the axis-direction connectivity functions

$p_c$9

are strictly decreasing in $p_c$0 when $p_c$1 is sufficiently close to $p_c$2 or $p_c$3, respectively. The proofs combine Ornstein-Zernike asymptotics for large $p_c$4 with polymer-expansion estimates for small $p_c$5 (Lima et al., 2015).

For supercritical Bernoulli bond percolation on $p_c$6 with $p_c$7 and $p_c$8 sufficiently close to $p_c$9, the finite connection function exhibits Ornstein-Zernike behaviour in every direction: $G=(V,E)$0 has asymptotics given by an exponential term $G=(V,E)$1 multiplied by a Gaussian prefactor of order $G=(V,E)$2, where $G=(V,E)$3 is an equivalent norm on $G=(V,E)$4, and the equi-decay surfaces $G=(V,E)$5 are locally analytic, strictly convex, and have positive Gaussian curvature (Campanino et al., 2011). This provides a directional large-deviation geometry for finite clusters embedded in a phase with an infinite cluster.

A one-dimensional defect can alter this asymptotic regime sharply. In subcritical percolation on $G=(V,E)$6 with parameter $G=(V,E)$7, if edges on the first coordinate axis are opened with probability $G=(V,E)$8, then the inverse correlation length

$G=(V,E)$9

stays equal to the homogeneous value $\theta(p)$0 for $\theta(p)$1, and becomes strictly smaller for $\theta(p)$2. The transition depends on dimension: $\theta(p)$3, whereas $\theta(p)$4 for $\theta(p)$5. In the pinned phase $\theta(p)$6,

$\theta(p)$7

so the polynomial Ornstein-Zernike correction disappears and the decay becomes purely exponential (Friedli et al., 2011).

The graph structure itself can change finite-connectivity decay in the supercritical phase. On bounded-degree graphs with $\theta(p)$8 and a bi-infinite geodesic, finite connectivity decays exponentially for $\theta(p)$9 sufficiently close to $\chi(p)=\mathbb{E}_p(|C(o)|)$0. By contrast, there are graphs with $\chi(p)=\mathbb{E}_p(|C(o)|)$1 and no bi-infinite geodesic for which the same quantity decays sub-exponentially, even polynomially, for $\chi(p)=\mathbb{E}_p(|C(o)|)$2 arbitrarily close to $\chi(p)=\mathbb{E}_p(|C(o)|)$3 (Alves et al., 2012). A recurrent misconception is therefore that highly supercritical finite connectivity must always be exponentially decaying; the graph-theoretic counterexamples show otherwise.

4. Extremal finite clusters and maximal diameter in the non-critical regime

A recent direction studies not only typical finite-cluster geometry but also extremal finite clusters inside large boxes. For $\chi(p)=\mathbb{E}_p(|C(o)|)$4, with $\chi(p)=\mathbb{E}_p(|C(o)|)$5, let $\chi(p)=\mathbb{E}_p(|C(o)|)$6 be the finite cluster containing $\chi(p)=\mathbb{E}_p(|C(o)|)$7, and define

$\chi(p)=\mathbb{E}_p(|C(o)|)$8

where the diameter is the maximal $\chi(p)=\mathbb{E}_p(|C(o)|)$9 span of the cluster in any coordinate direction (Kobayashi, 18 Dec 2025).

The fundamental rate is

$p$00

which also governs

$p$01

There exists $p$02 such that

$p$03

and the corresponding logarithmic scale is

$p$04

The main law of large numbers states that

$p$05

for every non-critical $p$06, under both free and zero boundary conditions (Kobayashi, 18 Dec 2025).

The same work derives a large deviation principle above the typical scale. For $p$07,

$p$08

so

$p$09

up to sub-polynomial corrections. If

$p$10

with $p$11, then for any $p$12 and large $p$13,

$p$14

and

$p$15

These results transfer earlier extremal-volume questions to extremal diameter and make the logarithmic scale of the largest finite cluster explicit (Kobayashi, 18 Dec 2025).

5. Inhomogeneity, reinforcement, anisotropy, and sprinkling

Bernoulli bond percolation is especially sensitive to low-dimensional inhomogeneities, but the effect depends on how the inhomogeneity is organized. On $p$16, with a reinforced random region $p$17 around the axis $p$18, the overlap model satisfies

$p$19

for all $p$20 and $p$21, while in the stack model non-percolation holds if $p$22. Under these moment conditions, the critical curve $p$23 is constant for $p$24: random one-dimensional reinforcement does not lower the threshold (Nascimento et al., 2024). This corrects a common intuition that any sufficiently favorable line defect should trigger percolation.

Other one-dimensional defects do lower the threshold. In brochette percolation on $p$25, a positive-density random set of vertical columns carries vertical edges of parameter $p$26, while all horizontal edges and vertical edges outside those columns have parameter $p$27. If $p$28, then $p$29 can be chosen strictly below $p$30 while the origin still percolates with positive probability. More precisely, for every $p$31 and $p$32, there exists $p$33 such that for almost every column environment,

$p$34

(Duminil-Copin et al., 2016).

Columnar disorder in $p$35 leads to a related strict-inequality phenomenon. If columns are removed independently with probability $p$36, and Bernoulli bond percolation of parameter $p$37 is then performed on the remaining graph, the critical curve $p$38 satisfies $p$39 for $p$40, but there exists $p$41 such that

$p$42

Thus the threshold remains uniformly strictly below $p$43 throughout the supercritical column-density regime (Hilário et al., 2020).

A complementary homogenization principle appears when a sparse Bernoulli perturbation is added to an already everywhere-percolating subgraph $p$44. If $p$45 is an independent $p$46-Bernoulli percolation and $p$47, then $p$48 is connected almost surely, $p$49 almost surely, and for every $p$50 a renormalized version of $p$51 stochastically dominates a $p$52-Bernoulli percolation (Benjamini et al., 2015). In this sense, sprinkling regularizes large-scale connectivity.

6. Analyticity, threshold identities, and related generalizations

Several structural questions concern the regularity of percolative observables as functions of the parameter. For Bernoulli bond percolation on $p$53, $p$54, the percolation density $p$55 is analytic on the entire supercritical interval $p$56, while the susceptibility $p$57 is analytic on $p$58 for all transitive short- or long-range models (Georgakopoulos et al., 2018). The same work also gives bond-percolation results for triangulations, including $p$59 for certain families satisfying the stated expansion or transience conditions (Georgakopoulos et al., 2018).

Threshold identification by cutsets is subtler than first-moment heuristics suggest. For a locally finite connected graph, the quantities $p$60 and $p$61 obtained from $p$62 always satisfy $p$63, but Kahn’s counterexample shows that equality can fail. The modified one-arm thresholds do recover the true critical point: $p$64 The key observation is that the one-arm expectation quantity also appears in the differential inequality of one-arm events, linking the problem to the Duminil-Copin–Tassion lemma (Tang, 2020). The controversy is therefore resolved by replacing raw cutset expectations with a better-adapted one-arm sum.

Bernoulli bond percolation also serves as a benchmark for broader connectivity models. Bernoulli hyper-edge percolation on $p$65 replaces edges by arbitrary finite hyper-edges, opened independently with probabilities

$p$66

Under the stated annulus-crossing and irreducibility conditions, the model has a non-trivial phase transition, uniqueness of the infinite cluster, and a Grimmett-Marstrand-type slab theorem in the supercritical regime (Chang, 2021). Conversely, comparisons with loop models show that percolation of open bonds is generally easier than the formation of infinite loops: on bounded-degree graphs,

$p$67

and on Galton-Watson trees with $p$68,

$p$69

These strict inequalities show that an infinite Bernoulli cluster is necessary but not sufficient for infinite-loop phenomena (1908.10213, Klippel et al., 5 Mar 2025).

Randomization of the bond parameters can preserve critical large-scale behaviour rather than destroy it. In the Bernoulli special case of the near-critical random bond FK model, if independent random edge parameters are centered around $p$70, then the quenched model almost surely looks critical at large scales; the summary further states that even non-degenerate i.i.d. parameters supported in $p$71 and centered at $p$72 yield large-scale crossing probabilities converging to their critical values in probability (Avérous et al., 10 Sep 2025). This suggests that, for Bernoulli percolation, centered quenched disorder can be irrelevant for large-scale criticality, even though deterministic deviations from $p$73 produce the usual near-critical crossover (Avérous et al., 10 Sep 2025).