• Corpus ID: 222177118

Long-range order in discrete spin systems

  title={Long-range order in discrete spin systems},
  author={Ron Peled and Yinon Spinka},
  journal={arXiv: Mathematical Physics},
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 threshold. The results characterize all periodic, maximal-pressure Gibbs states of the system. The results further apply in low dimensions provided that the lattice $\mathbb{Z}^d$ is replaced by $\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$ with $d_1\ge 2$ and $d=d_1+d_2$ sufficiently high, where $\mathbb{T… 

Figures from this paper

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
Finitary codings for gradient models and a new graphical representation for the six‐vertex model
The heart of the argument is to devise a suitable tree structure on the clusters of the underlying percolation process (associated to the graphical representation of the given model), which can be revealed piece-by-piece via exploration and deduce a volume-order large deviation estimate for the energy.
Uniqueness for the q-state antiferromagnetic Potts model on the regular tree
An elementary proof for the uniqueness regime of general q-state antiferromagnetic Potts model on the d-ary tree is presented and the exact exponential decay rate in all of the subcritical regime, and power law decay rate at the critical temperature is obtained.
From hard spheres to hard-core spins
A system of hard spheres exhibits physics that is controlled only by their density. This comes about because the interaction energy is either infinite or zero, so all allowed configurations have


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
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
Phase coexistence and torpid mixing in the 3-coloring model on ℤd
We show that for all sufficiently large $d$, the uniform proper 3-coloring model (in physics called the 3-state antiferromagnetic Potts model at zero temperature) on ${\mathbb Z}^d$ admits multiple
Delocalization of Uniform Graph Homomorphisms from $${\mathbb {Z}}^2$$ to $${\mathbb {Z}}$$
Graph homomorphisms from the $\mathbb{Z}^d$ lattice to $\mathbb{Z}$ are functions on $\mathbb{Z}^d$ whose gradients equal one in absolute value. These functions are the height functions corresponding
A Monotonicity Result for Hard-core and Widom-Rowlinson Models on Certain $d$-dimensional Lattices
For each $d\geq 2$, we give examples of $d$-dimensional periodic lattices on which the hard-core and Widom-Rowlinson models exhibit a phase transition which is monotonic, in the sense that there
On Phase Transition in the Hard-Core Model on ${\mathbb Z}^d$
The influence of the boundary on behaviour at the origin persists as the boundary recedes.
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.
Mixing properties of colorings of the $\mathbb{Z}^d$ lattice
We study and classify proper $q$-colorings of the $\mathbb Z^d$ lattice, identifying three regimes where different combinatorial behavior holds: (1) When $q\le d+1$, there exist frozen colorings,
Lectures on the Ising and Potts Models on the Hypercubic Lattice
Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions,
Bounding the Partition Function of Spin-Systems
An upper bound is obtained for the partition function (the normalizing constant which turns the assignment of weights on f into a probability distribution) in the case when $G$ is a regular bipartite graph.