Group-Invariant Percolation Processes

• Group-invariant percolation processes are random models on symmetric graphs, defined such that their probability laws remain unchanged under automorphism group actions.
• They utilize invariant measures, Poissonian constructions, and factor-of-iid methods to rigorously analyze connectivity, cluster structure, and critical phase transitions.
• These processes provide insights into practical phenomena, linking probabilistic behavior with algebraic properties like amenability, property (T), and the Haagerup property.

Group-invariant percolation processes are a central class of random geometric models studied on highly symmetric graphs, most commonly infinite, vertex-transitive graphs such as Cayley graphs of discrete groups, regular trees, and lattices. These processes are characterized by probabilistic laws invariant under a distinguished automorphism group, often the full automorphism group or the left (or right) translation action of the underlying group. Group invariance has profound consequences for the structural and probabilistic properties of percolation models, affecting connectivity, cluster structure, critical parameters, ergodic properties, and links with deep algebraic concepts such as amenability, Kazhdan’s property (T), and the Haagerup property.

1. Definition and Core Examples

A group-invariant percolation process on a graph $X$, typically a Cayley graph $X = \mathrm{Cay}(\Gamma, S)$ of a discrete group $\Gamma$ or a regular tree $T_d$, is a probability measure $\mathbb{P}$ on $\{0,1\}^E$ (bond percolation) or $\{0,1\}^V$ (site percolation) satisfying invariance under the action of $\Gamma$ or $\mathrm{Aut}(X)$. Namely,

$$\mathbb{P}(\omega \in A) = \mathbb{P}(g \cdot \omega \in A) \qquad \forall\, g \in \Gamma, \; A \subset \text{configuration space},$$

where $g \cdot \omega$ denotes the pushforward by graph automorphisms.

Canonical examples include:

• Bernoulli percolation: each edge (or site) is independently open with probability $p$, and the law is invariant under the full automorphism group.
• Poisson line or hyperplane processes: on $\mathbb{R}^d$, in the hyperbolic plane, or in trees, random geometric objects are distributed via an invariant Poisson process (e.g., Poisson lines in $T_3$ (Blanc, 2023)).
• Factor-of-iid processes: percolations realized as measurable, equivariant functions of an iid random labelling, resulting in processes invariant under automorphism (see (Rahman, 2014)).
• Gibbs and dependent percolation models: processes invariant under group action but possibly with complex dependencies (e.g., loop $O(n)$ models, random interlacements).

Group-invariant percolation underpins both classical models and a wide variety of recent constructions, including robust mechanisms for coexistence, retention under thinning, and interaction with algebraic or operator-algebraic invariants of the group.

2. Measures of Invariance and Construction

Group invariance constrains allowable random processes:

• Invariant measures on configuration space: For random objects such as lines in trees, there may exist a unique (up to scaling) locally finite Borel measure invariant under the automorphism group. For lines in the $3$-regular tree, this measure $\mu$ is uniquely determined by the property $\mu\{\ell\ni x,y\} = 2^{-d(x,y)}$ for any distinct vertex pair $x,y$ (Blanc, 2023).
• Poissonian constructions: Invariant Poisson processes are constructed using such invariant measures, leading to models where either the trace of the random objects (e.g., set of edges covered by Poisson lines) or the complement (“vacant set”) forms an invariant percolation.

For factor-of-iid percolation on $T_d$, each vertex receives an independent label, and the open/closed status is decided by a measurable function equivariant under the root-preserving automorphisms (Rahman, 2014).

In the context of group percolations on Cayley graphs, the invariance condition can be algebraically characterized and is compatible with random walks, heat kernels, and spectral properties of the group and graph (Mukherjee et al., 2023, Mukherjee et al., 2022).

3. Phase Transitions and Critical Behavior

Invariant percolation processes on transitive graphs display sharp phase transitions analogous to classical percolation, but with significant structural refinements:

• Existence of critical values: For Poisson line percolation in $T_3$, there is a critical intensity $\lambda_c = 4\ln2$ such that the vacant set percolates (contains an infinite component) for $\lambda < \lambda_c$ and all components are finite for $\lambda \ge \lambda_c$ (Blanc, 2023).
• First-passage and explosion transitions: When additional structure is introduced (e.g., marking lines with random speeds), further phase transitions emerge. For the speed-limited “road” model on $T_3$, the explosion transition occurs at Pareto parameter $\beta_c=2$: for $\beta<2$ a.s. infinite travel in finite time is possible, while for $\beta>2$ all infinite paths have infinite passage time.
• Nonamenable case and many clusters: On nonamenable Cayley graphs, invariant percolation can produce infinitely many infinite clusters, with their number and properties tightly constrained by symmetry and combinatorial arguments (Mukherjee et al., 2023, Glazman et al., 28 Aug 2025, Alami et al., 20 Dec 2025).
• Tree and high-degree behavior: For factor-of-iid percolation on $T_d$ with only finite clusters a.s., the density of open sites is asymptotically at most $(\log d)/d$ as $d\to\infty$, and this bound is tight (Rahman, 2014).
• Planar dichotomy: Invariant (tail-trivial, FKG-associative, stochastically dominated) processes on planar graphs a.s. admit either zero or infinitely many infinite clusters, but not a unique infinite cluster for $p\leq 1/2$ (Glazman et al., 28 Aug 2025).

These transitions are often analyzed via branching processes, mass-transport techniques, and entropy functionals exploiting invariance and ergodicity.

4. Algebraic and Analytical Characterizations

Group-invariant percolation processes serve as bridges between probabilistic geometry and group/algebraic theory:

• Amenability: The existence of invariant percolations with arbitrarily high density but only finite clusters characterizes amenability of the underlying group (Mukherjee et al., 2022). This is realized concretely by the BLPS theorem: $\Gamma$ is amenable if and only if, for each $\alpha<1$, there exists a $\Gamma$-invariant percolation of density at least $\alpha$ and only finite clusters.
• Haagerup property and property (T): $\Gamma$ has the Haagerup property if and only if for every $\alpha<1$ there exists a $\Gamma$-invariant percolation with mean degree above $\alpha \deg(o)$ and two-point function vanishing at infinity. Conversely, property (T) corresponds to the existence of a threshold $\alpha^*<1$, above which all $\Gamma$-invariant percolations necessarily have positive lower bounds for the connectivity function between distant vertices (Mukherjee et al., 2023).
• Operator-algebraic connections: Group-invariant percolations yield positive definite functions and Schur multipliers approaching the identity in operator topology, thus reflecting group amenability and other operator-algebraic properties (Mukherjee et al., 2022).

The interaction with $C^*$-algebraic invariants is structurally mediated via normalized indicator functions on clusters and their overlaps, with convergences in expectations characterizing algebraic properties.

5. Cluster Structure, Indistinguishability, and Robustness

Group invariance enforces rigid structure on the properties of clusters:

• Indistinguishability: For any group-invariant percolation on a Cayley graph producing multiple infinite clusters, these clusters are a.s. indistinguishable for all invariant cluster properties provided a mild “moving infinite branches” property holds, vastly generalizing earlier results that required finite energy and transience (Alami et al., 20 Dec 2025).
• Ergodicity and tail properties: Every invariant, ergodic percolation process on a Cayley graph has cluster subrelations ergodic for any non-tail property. Tail triviality becomes essential for the most robust indistinguishability conclusions.
• Robust coexistence and thinning: Certain translation-invariant percolation processes (e.g., a family on $\mathbb{Z}^2$ constructed by hierarchical grids) admit simultaneous infinite open and infinite closed clusters, and this coexistence remains after iid thinning (Hägström et al., 2010).
• Phase diagrams for dependent models: The loop $O(n)$ model on the hexagonal lattice, under translation-invariant Gibbs measures at $n\in[1,2]$, $x\in[1/\sqrt2,1]$, exhibits a.s. infinitely many macroscopic loops, connecting with percolation dichotomy results and critical surface predictions (Glazman et al., 28 Aug 2025).

6. Extensions, Methodological Innovations, and Open Questions

Recent work extends the framework of group-invariant percolation:

• Beyond classical settings: Invariant percolation has been extended to hyperbolic spaces, spaces with measured walls, Cayley graphs of infinitely generated groups, and to processes determined by point processes on families of “obstacles” with invariant measures (Georgakopoulos et al., 2017, Blanc, 2023, Mukherjee et al., 2023).
• Growth processes and interacting particle systems: Many interacting growth models, including truncated or competing branching random walks, encode survival or coexistence through the existence of infinite clusters in suitably invariant percolation representations (Müller, 2013).
• Critical behavior on infinitely generated groups: In certain infinitely generated groups that are not quasi-isometric to $\mathbb{Z}$, all clusters are finite for any parameter value, showing that naive analogues of the $p_c<1$ paradigm can fail dramatically (Georgakopoulos et al., 2017).
• Entropy-based and mass-transport proofs: Variational principles, entropy inequalities, and mass-transport are primary analytic tools for proving extremal and rigidity statements about invariant percolations, especially in factor-of-iid and nonamenable cases (Rahman, 2014, Müller, 2013).

Challenges remain in classifying phase transitions for hereditary just-infinite groups, determining precise critical exponents in dependent models, and extending indistinguishability to even broader classes of random processes.

7. Summary Table: Invariant Percolation Properties

Graph/Group Setting	Invariant Percolation Behavior	Structural Theorem/Paper
$T_3$ (3-reg. tree)	Sharp vacant set percolation at $\lambda_c$	(Blanc, 2023)
$T_d$, factor-of-iid	Max. finite-cluster density $\sim\frac{\log d}{d}$	(Rahman, 2014)
Cayley graph, amenable	Arbitrarily dense, finite-cluster percolations	(Mukherjee et al., 2022)
Property (T) group	Long-range order above threshold $\alpha^*$	(Mukherjee et al., 2023)
Nonamenable planar	Only $0$ or $\infty$ infinite clusters for $p\le1/2$	(Glazman et al., 28 Aug 2025)
$\mathbb{Z}^2$	Translation-invariant processes with coexistence	(Hägström et al., 2010)
Infinitely generated	All clusters finite for all $A>0$	(Georgakopoulos et al., 2017)

This table summarizes key behaviors established for group-invariant percolation across several canonical settings, with direct citation to the corresponding theoretical developments.