# On non-uniqueness of percolation on nonamenable Cayley graphs *

@article{Pak2000OnNO, title={On non-uniqueness of percolation on nonamenable Cayley graphs *}, author={Igor Pak and Tatiana Smirnova-Nagnibeda}, journal={Comptes Rendus De L Academie Des Sciences Serie I-mathematique}, year={2000}, volume={330}, pages={495-500} }

## 76 Citations

### Small spectral radius and percolation constants on non-amenable Cayley graphs

- Mathematics
- 2012

Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated nonamenable group Gamma, does there exist a generating set S…

### Percolation on nonunimodular transitive graphs

- Mathematics
- 2006

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 critical…

### Percolation on Hyperbolic Graphs

- MathematicsGeometric and Functional Analysis
- 2019

We prove that Bernoulli bond percolation on any nonamenable, Gromov hyperbolic, quasi-transitive graph has a phase in which there are infinitely many infinite clusters, verifying a well-known…

### Percolation in the hyperbolic space: non-uniqueness phase and fibrous clusters

- Mathematics
- 2015

In this thesis, I am going to consider Bernoulli percolation on graphs admitting vertex-transitive actions of groups of isometries of d-dimensional hyperbolic spaces H^d. In the first chapter, I give…

### Non-amenable Cayley graphs of high girth have $p_c < p_u$ and mean-field exponents

- Mathematics
- 2012

In this note we show that percolation on non-amenable Cayley graphs of high girth has a phase of non-uniqueness, i.e., $p_c< p_u$. Furthermore, we show that percolation and self-avoiding walk on such…

### Invariant percolation and harmonic Dirichlet functions

- Mathematics
- 2004

Abstract.The main goal of this paper is to answer Question 1.10 and settle Conjecture 1.11 of Benjamini–Lyons–Schramm [BenLS] relating harmonic Dirichlet functions on a graph to those on the infinite…

### Non-unitarisable representations and random forests

- Mathematics
- 2008

We establish a connection between Dixmier's unitarisability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is non-unitarisable if its first…

### Clusters in middle-phase percolation on hyperbolic plane

- Mathematics
- 2011

I consider p-Bernoulli bond percolation on graphs of vertex-transitive tilings of the hyperbolic plane with finite sided faces (or, equivalently, on transitive, nonamenable, planar graphs with one…

### Coarse geometry and randomness

- Mathematics
- 2013

Isoperimetry and expansions in graphs.- Several metric notions.- The hyperbolic plane and hyperbolic graphs.- More on the structure of vertex transitive graphs.- Percolation on graphs.- Local limits…

### Non-uniqueness phase of Bernoulli percolation on reflection groups for some polyhedra in H^3

- Mathematics
- 2013

In the present paper I consider Cayley graphs of reflection groups of finite-sided Coxeter polyhedra in 3-dimensional hyperbolic space H^3, with standard sets of generators. As the main result, I…

## References

SHOWING 1-10 OF 12 REFERENCES

### Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

- Mathematics
- 1999

Abstract. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we…

### Density and uniqueness in percolation

- Mathematics
- 1989

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 has…

### Percolation Beyond $Z^d$, Many Questions And a Few Answers

- Mathematics
- 1996

A comprehensive study of percolation in a more general context than the usual $Z^d$ setting is proposed, with particular focus on Cayley graphs, almost transitive graphs, and planar graphs. Results…

### Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation

- Mathematics
- 1987

For independent translation-invariant irreducible percolation models, it is proved that the infinite cluster, when it exists, must be unique. The proof is based on the convexity (or almost convexity)…

### 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…

### Phase transitions on nonamenable graphs

- Mathematics, Physics
- 2000

We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees.

### Percolation in ∞ + 1 dimensions, in: Disorder in Physical Systems

- 1990

### Percolation, Springer-Verlag

- New York,
- 1989