An isoperimetric inequality on the discrete torus
SIAM Journal on Discrete Mathematics
Fractional v. integral covers in hypergraphs of bounded edge size
Journal of Combinatorial Theory Series A
Asymptotics of the list-chromatic index for multigraphs
Random Structures & Algorithms
An Entropy Approach to the Hard-Core Model on Bipartite Graphs
Combinatorics, Probability and Computing
Slow mixing of glauber dynamics via topological obstructions
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Slow mixing of Glauber dynamics for the hard-core model on regular bipartite graphs
Random Structures & Algorithms
Torpid mixing of local Markov chains on 3-colorings of the discrete torus
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Slow Mixing of Markov Chains Using Fault Lines and Fat Contours
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
The 1-vertex transfer matrix and accurate estimation of channel capacity
IEEE Transactions on Information Theory
A threshold phenomenon for random independent sets in the discrete hypercube
Combinatorics, Probability and Computing
Journal of Combinatorial Theory Series B
Independence entropy of Z^d-shift spaces
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Hi-index | 0.06 |
It is shown that the hard-core model on ${{\mathbb Z}}^d$ exhibits a phase transition at activities above some function $\lambda(d)$ which tends to zero as $d\rightarrow \infty$. More precisely, consider the usual nearest neighbour graph on ${{\mathbb Z}}^d$, and write ${\cal E}$ and ${\cal O}$ for the sets of even and odd vertices (defined in the obvious way). Set $${\cal G}L_M={\cal G}L_M^d =\{z\in{{\mathbb Z}}^d:\|z\|_{\infty}\leq M\},\quad \partial^{\star} {\cal G}L_M =\{z\in{{\mathbb Z}}^d:\|z\|_{\infty}= M\},$$ and write ${\cal I}({\cal G}L_M)$ for the collection of independent sets (sets of vertices spanning no edges) in ${\cal G}L_M$. For $\lambda0$ let ${\bf I}$ be chosen from ${\cal I}({\cal G}L_M)$ with $\Pr({\bf I}=I) \propto \lambda^{|I|}$.Theorem There is a constant $C$ such that if $\lambda Cd^{-1/4}\log^{3/4}d$, then $$\lim_{M\rightarrow\infty}\Pr(\underline{0}\in{\bf I}|{\bf I}\supseteq \partial^{\star} {\cal G}L_M\cap {\cal E})~ \lim_{M\rightarrow\infty}\Pr(\underline{0}\in{\bf I}| {\bf I}\supseteq \partial^{\star} {\cal G}L_M\cap {\cal O}).$$ Thus, roughly speaking, the influence of the boundary on behaviour at the origin persists as the boundary recedes.