Some intersection theorems for ordered sets and graphs
Journal of Combinatorial Theory Series A
Conductance and the rapid mixing property for Markov chains: the approximation of permanent resolved
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Graph homomorphisms and phase transitions
Journal of Combinatorial Theory Series B
On random graph homomorphisms into Z
Journal of Combinatorial Theory Series B
On Counting Independent Sets in Sparse Graphs
SIAM Journal on Computing
Torpid Mixing of Some Monte Carlo Markov Chain Algorithms in Statistical Physics
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
An Entropy Approach to the Hard-Core Model on Bipartite Graphs
Combinatorics, Probability and Computing
On Phase Transition in the Hard-Core Model on ${\mathbb Z}^d$
Combinatorics, Probability and Computing
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
Sampling independent sets in the discrete torus
Random Structures & Algorithms
A threshold phenomenon for random independent sets in the discrete hypercube
Combinatorics, Probability and Computing
A very simple algorithm for estimating the number of k‐colorings of a low‐degree graph
Random Structures & Algorithms
Journal of Combinatorial Theory Series B
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For graphs G and H, an H-coloring of G is a function from the vertices of G to the vertices of H that preserves adjacency. H-colorings encode graph theory notions such as independent sets and proper colorings, and are a natural setting for the study of hard-constraint models in statistical physics. We study the set of H-colorings of the even discrete torus Z"m^d, the graph on vertex set {0,...,m-1}^d (m even) with two strings adjacent if they differ by 1 (mod m) on one coordinate and agree on all others. This is a bipartite graph, with bipartition classes E and O. In the case m=2 the even discrete torus is the discrete hypercube or Hamming cube Q"d, the usual nearest neighbor graph on {0,1}^d. We obtain, for any H and fixed m, a structural characterization of the space of H-colorings of Z"m^d. We show that it may be partitioned into an exceptional subset of negligible size (as d grows) and a collection of subsets indexed by certain pairs (A,B)@?V(H)^2, with each H-coloring in the subset indexed by (A,B) having all but a vanishing proportion of vertices from E mapped to vertices from A, and all but a vanishing proportion of vertices from O mapped to vertices from B. This implies a long-range correlation phenomenon for uniformly chosen H-colorings of Z"m^d with m fixed and d growing. The special pairs (A,B)@?V(H)^2 are characterized by every vertex in A being adjacent to every vertex in B, and having |A||B| maximal subject to this condition. Our main technical result is an upper bound on the probability, for an arbitrary edge uv of Z"m^d, that in a uniformly chosen H-coloring f of Z"m^d the pair ({f(w):w@?N"u},{f(z):z@?N"v}) is not one of these special pairs (where N"@? indicates neighborhood). Our proof proceeds through an analysis of the entropy of f, and extends an approach of Kahn, who had considered the case of m=2 and H a doubly infinite path. All our results generalize to a natural weighted model of H-colorings.