• 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}
}
• Published 14 June 2012
• Physics, Computer Science
• 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 $d\rightarrow \infty$
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## 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
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
• Mathematics
SIAM J. Discret. Math.
• 1990
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
• Mathematics
• 1999
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 ?
• Mathematics
• 1982
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 number of antichains in multilevelled ranked posets
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