On non-uniqueness of percolation on nonamenable Cayley graphs *

  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},

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

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

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

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

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

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

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

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

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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

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

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



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

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

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

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

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

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

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.

Full Banach Mean Values on Countable groups.

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