# 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 $d\rightarrow \infty$

## 39 Citations

### Odd cutsets and the hard-core model on Z^d

- Computer Science
- 2011

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

- Mathematics
- 2015

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

- MathematicsSIAM J. Discret. Math.
- 2011

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

- Mathematics
- 2012

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

- Mathematics, Computer ScienceSODA
- 2019

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

- Mathematics
- 2020

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

- Computer ScienceSTOC
- 2019

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

- MathematicsJournal of the European Mathematical Society
- 2019

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

- Mathematics
- 2017

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

- Mathematics, Computer ScienceProbability Theory and Related Fields
- 2018

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.

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