• Corpus ID: 222177118

Long-range order in discrete spin systems

@article{Peled2020LongrangeOI,
  title={Long-range order in discrete spin systems},
  author={Ron Peled and Yinon Spinka},
  journal={arXiv: Mathematical Physics},
  year={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 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… 
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References

SHOWING 1-10 OF 92 REFERENCES
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 ${\mathbb Z}^d$
TLDR
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
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.
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
TLDR
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.
...
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