# Group-invariant Percolation on Graphs

@article{Benjamini1999GroupinvariantPO, title={Group-invariant Percolation on Graphs}, author={I. Benjamini and R. Lyons and Y. Peres and O. Schramm}, journal={Geometric \& Functional Analysis GAFA}, year={1999}, volume={9}, pages={29-66} }

Abstract. Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new masstransport technique that has been occasionally used elsewhere and is developed further here.¶ Perhaps surprisingly, these investigations of group-invariant… Expand

#### 214 Citations

Site Percolation on Planar Graphs

- Mathematics
- 2020

We prove that for a non-amenable, locally finite, connected, transitive, planar graph with one end, any automorphism invariant site percolation on the graph does not have exactly 1 infinite 1-cluster… Expand

Percolation Perturbations inPotential Theory and Random

- 1998

We show that on a Cayley graph of a nonamenable group, a.s. the innnite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive… Expand

Invariant percolation on trees and the mass-transport method

- Mathematics
- 1999

In bond percolation on an in nite locally nite graph G = (V;E), each edge is randomly assigned value 0 (absent) or 1 (present) according to some probability measure on f0; 1g. One then studies… Expand

Critical percolation on certain nonunimodular graphs

- Mathematics
- 2006

An important conjecture in percolation theory is that almost sure- ly no infinite cluster exists in critical percolation on any transitive graph for which the critical probability is less than 1.… Expand

Invariant random graphs with iid degrees in a general geography

- Mathematics
- 2009

Let D be a non-negative integer-valued random variable and let G = (V, E) be an infinite transitive finite-degree graph. Continuing the work of Deijfen and Meester (Adv Appl Probab 38:287–298) and… Expand

Random walks on random subgraphs of transitive graphs

In the past few years, random walks on percolative clusters of the Euclidean lattice Z have been investigated with respect to the question of their return probability [5, 10, 39, 19, 28]. At the same… Expand

Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs

- Mathematics
- 2001

Abstract: We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs.It is shown that on graphs with edge-isoperimetric… Expand

An Interlacing Technique for Spectra of Random Walks and Its Application to Finite Percolation Clusters

- Mathematics
- 2010

A comparison technique for random walks on finite graphs is introduced, using the well-known interlacing method. It yields improved return probability bounds. A key feature is the incorporation of… Expand

Percolation and isoperimetry on roughly transitive graphs

- Mathematics
- 2015

In this paper we study percolation on a roughly transitive graph G with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that p_c < 1, or in other… Expand

Percolation on Transitive Graphs as a Coalescent Process : Relentless Merging Followed by Simultaneous

- 1998

Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter pc ∈ [0, 1] for the existence of infinite open clusters. Recently, it has been… Expand

#### References

SHOWING 1-10 OF 104 REFERENCES

Infinite clusters in dependent automorphism invariant percolation on trees

- Mathematics
- 1997

We study dependent bond percolation on the homogeneous tree T n of order n ≥ 2 under the assumption of automorphism invariance. Excluding a trivial case, we find that the number of infinite clusters… Expand

Percolation Perturbations in Potential Theory and Random Walks

- Mathematics, Physics
- 1998

We show that on a Cayley graph of a nonamenable group, almost surely the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has… Expand

Difference equations, isoperimetric inequality and transience of certain random walks

- Mathematics
- 1984

The difference Laplacian on a square lattice in Rn has been stud- ied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many… Expand

On the norms of group-invariant transition operators on graphs

- Mathematics
- 1992

In this paper we consider reversible random walks on an infinite grapin, invariant under the action of a closed subgroup of automorphisms which acts with a finite number of orbits on the vertex-set.… Expand

Percolation and minimal spanning forests in infinite graphs

- Mathematics
- 1995

The structure of a spanning forest that generalizes the minimal spanning tree is considered for infinite graphs with a value f(b) attached to each bond b. Of particular interest are stationary random… Expand

Percolation in the Hyperbolic Plane Extended Abstract

The Voronoi model for percolation in H 2. Percolation has been studied extensively in R d and in lattices of R d. In 1], we have proposed some conjectures about percolation in quite general settings.… Expand

Aperiodic tilings, positive scalar curvature, and amenability of spaces

- Mathematics
- 1992

The object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity. Manifolds of this sort arise as leaves of foliations of compact manifolds and… Expand

Amenability, Kazhdan’s property and percolation for trees, groups and equivalence relations

- Mathematics
- 1991

We prove amenability for a broad class of equivalence relations which have trees associated to the equivalence classes. The proof makes crucial use of percolation on trees. We also discuss related… Expand

Uniform and minimal essential spanning forests on trees

- Computer Science
- Random Struct. Algorithms
- 1998

The uniform essential spanning forest admits a simple description in terms of a Galton]Watson process and also arises as a limit of random-cluster measures. Expand

Random walks on graphs with a strong isoperimetric property

- Mathematics
- 1988

A random walk on a graph is a Markov chain whose state space consists of the vertices of the graph and where transitions are only allowed along the edges. We study (strongly) reversible random walks… Expand