Results 1  10
of
26
Dynamical stability of percolation for some interacting particle systems and fflmovability
, 2004
"... In this paper we will investigate dynamic stability of percolation for the stochastic Ising model and the contact process. We also introduce the notion of downward and upward εmovability which will be a key tool for our analysis. 1. Introduction. Consider ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
In this paper we will investigate dynamic stability of percolation for the stochastic Ising model and the contact process. We also introduce the notion of downward and upward εmovability which will be a key tool for our analysis. 1. Introduction. Consider
The Ising Model on Diluted Graphs and Strong Amenability
, 1999
"... Say that a graph has persistent transition if the Ising model on the graph can exhibit a phase transition (nonuniqueness of Gibbs measures) in the presence of a nonzero external field. We show that for nonamenable graphs, for Bernoulli percolation with p close to 1, all the infinite clusters have pe ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Say that a graph has persistent transition if the Ising model on the graph can exhibit a phase transition (nonuniqueness of Gibbs measures) in the presence of a nonzero external field. We show that for nonamenable graphs, for Bernoulli percolation with p close to 1, all the infinite clusters have persistent transition. On the other hand, we show that for transitive amenable graphs, the infinite clusters for any stationary percolation do not have persistent transition. This extends a result of Georgii for the cubic lattice. A geometric consequence of this latter fact is that the infinite clusters are strongly amenable (i.e., their anchored Cheeger constant is 0). Finally we show that the critical temperature for the Ising model with no external field on the infinite clusters of Bernoulli percolation with parameter p, on an arbitrary bounded degree graph, is a continuous function of p.
GIBBS RANDOM FIELDS WITH UNBOUNDED SPINS ON UNBOUNDED DEGREE GRAPHS
, 904
"... Abstract. Gibbs random fields corresponding to systems of realvalued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs with a certain summability property are constructed. It is proven that the set of tempered Gibbs random fields is nonvo ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
Abstract. Gibbs random fields corresponding to systems of realvalued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs with a certain summability property are constructed. It is proven that the set of tempered Gibbs random fields is nonvoid and weakly compact, and that they obey uniform exponential integrability estimates. In the second part of the paper, a class of graphs is described in which the mentioned summability is obtained as a consequence of a property, by virtue of which vertices of large degree are located at large distances from each other. The latter is a stronger version of a metric property, introduced in [Bassalygo, L. A. and Dobrushin, R. L. (1986). Uniqueness of a Gibbs field with a random potential—an elementary approach. Theory Probab. Appl. 31 572–589]. 1. Introduction and paper
Phase transition for the Ising model on the Critical Lorentzian triangulation
, 2008
"... Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disag ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disagreement percolation method and on a variant of Peierls method. The critical temperature is shown to be constant a.s.
Percolation on Grigorchuk groups
 Comm. Algebra
"... Let pc(G) be the critical probability of the site percolation on the Cayley graph of group G. In [2] of Benjamini and Schramm conjectured thatpc < 1, given the group is infinite and not a finite extension of Z. The conjecture was proved earlier for groups of polynomial and exponential growth and ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
(Show Context)
Let pc(G) be the critical probability of the site percolation on the Cayley graph of group G. In [2] of Benjamini and Schramm conjectured thatpc < 1, given the group is infinite and not a finite extension of Z. The conjecture was proved earlier for groups of polynomial and exponential growth and remains open for groups of intermediate growth. In this note we prove the conjecture for a special class of Grigorchuk groups, which is a special class of groups of intermediate growth. The proof is based on an algebraic construction. No previous knowledge of percolation is assumed. 1
On the Cluster Size Distribution for Percolation on Some General Graphs
, 2010
"... We show that for any Cayley graph, the probability (at any p) that the cluster of the origin has size n decays at a welldefined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical regime with the amenability/nonamenability of the underlying gra ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
We show that for any Cayley graph, the probability (at any p) that the cluster of the origin has size n decays at a welldefined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical regime with the amenability/nonamenability of the underlying graph.
Weak survival for branching random walks on graphs
, 2007
"... We study weak and strong survival for branching random walks on multigraphs. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometrical parameter of the multigraph. For a large class of multigraphs we prove that, at the ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
(Show Context)
We study weak and strong survival for branching random walks on multigraphs. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometrical parameter of the multigraph. For a large class of multigraphs we prove that, at the weak critical value, the process dies out globally almost surely. Moreover for the same class we prove that the existence of a pure weak phase is equivalent to nonamenability; this improves a result of Stacey [14].
A monotonicity result for hardcore and WidomRowlinson models on certain ddimensional lattices
, 2001
"... For each d 2, we give examples of ddimensional periodic lattices on which the hardcore and WidomRowlinson models exhibit a phase transition which is monotonic, in the sense that there exists a critical value c for the activity parameter , such that there is a unique Gibbs measure (resp. multi ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
For each d 2, we give examples of ddimensional periodic lattices on which the hardcore and WidomRowlinson models exhibit a phase transition which is monotonic, in the sense that there exists a critical value c for the activity parameter , such that there is a unique Gibbs measure (resp. multiple Gibbs measures) whenever ! c (resp. ? c ). This contrasts with earlier examples of such lattices, where the phase transition failed to be monotonic. The case of the cubic lattice Z d remains an open problem. 1
Rlocal Delaunay inhibition Model
, 2006
"... Let us consider the local specification system of Gibbs point process with inhibition pairwise interaction acting on some Delaunay subgraph specifically not containing the edges of Delaunay triangles with circumscribed circle of radius greater than some fixed positive real value R. Even if we think ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Let us consider the local specification system of Gibbs point process with inhibition pairwise interaction acting on some Delaunay subgraph specifically not containing the edges of Delaunay triangles with circumscribed circle of radius greater than some fixed positive real value R. Even if we think that there exists at least a stationary Gibbs state associated to such system, we do not know yet how to prove it mainly due to some uncontrolled “negative ” contribution in the expression of the local energy needed to insert any number of points in some large enough empty region of the space. This is solved by introducing some subgraph, called the Rlocal Delaunay graph, which is a slight but tailored modification of the previous one. This kind of model does not inherit the local stability property but satisfies some new extension called Rlocal stability. This weakened property combined with the local property provides the existence of Gibbs state.