Infinite clusters in dependent automorphism invariant percolation on trees

@article{Hggstrm1997InfiniteCI,
  title={Infinite clusters in dependent automorphism invariant percolation on trees},
  author={Olle H{\"a}ggstr{\"o}m},
  journal={Annals of Probability},
  year={1997},
  volume={25},
  pages={1423-1436}
}
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 a.s. is either 0 or ∞. Furthermore, each infinite cluster a.s. has either 1, 2 or infinitely many topological ends, and infinite clusters with infinitely many topological ends have a.s. a branching number greater than 1. We also show that if the marginal probability that a single edge is open is at… Expand
Critical Percolation on any Nonamenable Group has no Infinite Clusters
We show that independent percolation on any Cayley graph of a nonamenable group has no infinite components at the critical parameter. This result was obtained by the present authors earlier as aExpand
Extremal point of infinite clusters in stationary percolation
It is well known that in stationary percolation, an infinite component cannot have a finite number of extremal points in a certain direction. In this note, we investigate whether or not an infiniteExpand
Percolation Perturbations in Potential Theory and Random Walks
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 hasExpand
Group-Invariant Percolation on Graphs by Itai Benjamini
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, suchExpand
Percolation on nonamenable products at the uniqueness threshold
Let X and Y be infinite quasi-transitive graphs, such that the automorphism group of X is not amenable. For i.i.d. percolation on the direct product X×Y , we show that the set of retention parametersExpand
Indistinguishability of Percolation Clusters
We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies thatExpand
Percolation on nonunimodular transitive graphs
We extend some of the fundamental results about percolation on unimodular nonamenable graphs to nonunimodular graphs. We show that they cannot have infinitely many infinite clusters at criticalExpand
Percolation on Transitive Graphs as a Coalescent Process : Relentless Merging Followed by Simultaneous
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 beenExpand
Group-invariant Percolation on Graphs
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 processesExpand
Invariant percolation on trees and the mass-transport method
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 studiesExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 16 REFERENCES
Topological and metric properties of infinite clusters in stationary two-dimensional site percolation
A dependent percolation model is a random coloring of the two-lattice. It is assumed that there are a finite number of colors and that the coloring is translation invariant. Each color defines aExpand
Density and uniqueness in percolation
Two results on site percolation on thed-dimensional lattice,d≧1 arbitrary, are presented. In the first theorem, we show that for stationary underlying probability measures, each infinite cluster hasExpand
Percolation and minimal spanning forests in infinite graphs
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 randomExpand
Tree-indexed random walks on groups and first passage percolation
SummarySuppose that i.i.d. random variables are attached to the edges of an infinite tree. When the tree is large enough, the partial sumsSσ along some of its infinite paths will exhibit behaviorExpand
Correction: Random walk in a random environment and first-passage percolation on trees
We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is aExpand
Infinite clusters in percolation models
The qualitative nature of infinite clusters in percolation models is investigated. The results, which apply to both independent and correlated percolation in any dimension, concern the number andExpand
Choosing a Spanning Tree for the Integer Lattice Uniformly
Consider the nearest neighbor graph for the integer lattice Zd in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphsExpand
Random Walks and Percolation on Trees
There is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree. It equals the exponential of the Hausdorff dimension of the boundary in an appropriateExpand
The random-cluster model on a homogeneous tree
SummaryThe random-cluster model on a homogeneous tree is defined and studied. It is shown that for 1≦q≦2, the percolation probability in the maximal random-cluster measure is continuous inp, whileExpand
Amenability, Kazhdan’s property and percolation for trees, groups and equivalence relations
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 relatedExpand
...
1
2
...