• Corpus ID: 3180920

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

  title={On phase transition in the hard-core model on \$\{\bf Z\}^d\$},
  author={David J. Galvin and Jeff Kahn},
  journal={arXiv: Combinatorics},
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

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.

Uniqueness of the translation-invariant Gibbs measure for Hard-Core models with four-states on the Cayley tree

We consider fertile HC-models with four-states and the parametre activity on a Cayley tree. It is known that three types of such models exist. For each of these models we prove uniqueness of the

The Multistate Hard Core Model on a Regular Tree

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.

Algorithms for #BIS-hard problems on expander graphs

An FPTAS and an efficient sampling algorithm are given and efficient counting and sampling algorithms are found for proper $q$-colorings of random $\Delta$-regular bipartite graphs when q is sufficiently small as a function of $\Delta$.

Long-range order in discrete spin systems

We establish long-range order for discrete nearest-neighbor spin systems on $\mathbb{Z}^d$ satisfying a certain symmetry assumption, when the dimension $d$ is higher than an explicitly described

Algorithmic Pirogov–Sinai theory

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

A condition for long-range order in discrete spin systems with application to the antiferromagnetic Potts model

We give a general condition for a discrete spin system with nearest-neighbor interactions on the $\mathbb{Z}^d$ lattice to exhibit long-range order. The condition is applicable to systems with

Percolation games, probabilistic cellular automata, and the hard-core model

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.



An Entropy Approach to the Hard-Core Model on Bipartite Graphs

  • J. Kahn
  • Mathematics
    Combinatorics, Probability and Computing
  • 2001
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.

An Isoperimetric Inequality on the Discrete Torus

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

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

Two Combinatorial Covering Theorems

  • S. Stein
  • Mathematics
    J. Comb. Theory, Ser. A
  • 1974

On Dedekind’s problem: The number of monotone Boolean functions

with an = ce~nli, /3„ = e'(Iog w)/«1'2.The number \p(n) is equal to the number of ideals, or of antichains,or of monotone increasing functions into 0 and 1 definable on thelattice of subsets of an