• Corpus ID: 3180920

On phase transition in the hard-core model on ${\bf Z}^d$

@article{Galvin2012OnPT,
  title={On phase transition in the hard-core model on \$\{\bf Z\}^d\$},
  author={David J. Galvin and Jeff Kahn},
  journal={arXiv: Combinatorics},
  year={2012}
}
It is shown that the hard-core model on ${\bf Z}^d$ exhibits a phase transition at activities above some function $\lambda(d)$ which tends to zero as $d\rightarrow \infty$ 
Odd cutsets and the hard-core model on Z^d
TLDR
It is proved that when the intensity parameter exceeds Cd^{-1/3}(log d)^2, the model exhibits multiple hard-core measures, thus improving the previous bound of Cd −1/4( log d)^{3/4} given by Galvin and Kahn.
The Multistate Hard Core Model on a Regular Tree
TLDR
This work considers this generalization of the hard core model as an idealized model of multicasting in communication networks when G is an infinite rooted b-ary tree and proves rigorously that the probability of a configuration σ∶V(G)→{0,…,C} is proportional to λ∑v∈V (G)σ(v).
On four state Hard Core Models on the Cayley Tree
We consider a nearest-neighbor four state hard-core (HC) model on the homogeneous Cayley tree of order $k$. The Hamiltonian of the model is considered on a set of "admissible" configurations.
Algorithmic Pirogov–Sinai theory
TLDR
An FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of Z d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus at sufficiently low temperature are developed.
Long-range order in the 3-state antiferromagnetic Potts model in high dimensions
We prove the existence of long-range order for the 3-state Potts antiferromagnet at low temperature on $\mathbb{Z}^d$ for sufficiently large $d$. In particular, we show the existence of six extremal
Percolation games, probabilistic cellular automata, and the hard-core model
TLDR
It is proved that these PCA are ergodic, and correspondingly that the game on Z2 has no draws, and it is shown that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions.
Rigidity of proper colorings of $\mathbb{Z}^d$
A proper $q$-coloring of a domain in $\mathbb{Z}^d$ is a function assigning one of $q$ colors to each vertex of the domain such that adjacent vertices are colored differently. Sampling a proper
The Uniqueness of the Translation-invariant Gibbs Measure for Four State HC-models on a Cayley Tree
We consider fertile Hard-Core (HC) models with activity parameter � > 0 and four states on the Cayley tree of order two. It is known that there are three types of such models. In this paper for each
Efficient algorithms for the Potts model on small-set expanders
TLDR
An approximation algorithm for the partition function of the ferromagnetic Potts model on graphs with a small-set expansion condition is developed and a graph partitioning algorithm with expansion and minimum degree conditions on the subgraphs induced by each part is given.
The Number of Maximal Independent Sets in the Hamming Cube
Let $Q_n$ be the $n$-dimensional Hamming cube and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically \[2n2^{N/4},\] as was conjectured by Ilinca and the first
...
...

References

SHOWING 1-10 OF 36 REFERENCES
An Entropy Approach to the Hard-Core Model on Bipartite Graphs
  • J. Kahn
  • Mathematics
    Combinatorics, Probability and Computing
  • 2001
TLDR
Results obtained include rather precise bounds on occupation probabilities; a ‘phase transition’ statement for Hamming cubes; and an exact upper bound on the number of independent sets in an n-regular bipartite graph on a given number of vertices.
Percolation and the hard-core lattice gas model
An Isoperimetric Inequality on the Discrete Torus
TLDR
The main aim of this paper is to give a best possible lower bound for A, the set of vertices of $\mathbb{Z}_k^n $ within distance t of A, for even values of k.
Nonmonotonic Behavior in Hard-Core and Widom–Rowlinson Models
We give two examples of nonmonotonic behavior in symmetric systems exhibiting more than one critical point at which spontanoous symmetry breaking appears or disappears. The two systems are the
Gibbs Measures and Phase Transitions
TLDR
This comprehensive monograph covers in depth a broad range of topics in the mathematical theory of phase transition in statistical mechanics and serves both as an introductory text and as a reference for the expert.
Entropy, independent sets and antichains: A new approach to Dedekind's problem
For n-regular, N-vertex bipartite graphs with bipartition A U B, a precise bound is given for the sum over independent sets I of the quantity μ |I ∩ A| λ |I ∩ B| , (In other language, this is
Percolation ?
572 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Percolation is a simple probabilistic model which exhibits a phase transition (as we explain below). The simplest version takes place on Z2, which we view
On the ratio of optimal integral and fractional covers
On the number of antichains in multilevelled ranked posets
TLDR
The asymptotics of the number of antichains in some multilevelled ranked posets is obtained, and the asymPTotics of a number of monotone Boolean functions and Boolean functions possessing the property < A^ > are obtained.
Two Combinatorial Covering Theorems
  • S. Stein
  • Mathematics
    J. Comb. Theory, Ser. A
  • 1974
...
...