It is shown that the hard-core model on Z d exhibits a phase transition at activities above some function λ(d) which tends to zero as d → ∞. More precisely, consider the usual nearest neighbour graph on Z d , and write E and O for the sets of even and odd vertices (defined in the obvious way). Setwrite I(Λ M ) for the collection of independent sets (sets of vertices spanning no edges) in Λ M . For λ > 0 let I be chosen from I(Λ M ) with Pr(I = I) ∝ λ |I| .Theorem. There is a constant C such that if λ > Cd −1/4 log 3/4 d, thenThus, roughly speaking, the influence of the boundary on behaviour at the origin persists as the boundary recedes.

For given graphs G and H, let |Hom(G, H)| denote the set of graph homomorphisms from G to H. We show that for any finite, n-regular, bipartite graph G and any finite graph H (perhaps with loops), |Hom(G, H)| is maximum when G is a disjoint union of K n,n 's. This generalizes a result of J. Kahn on the number of independent sets in a regular bipartite graph. We also give the asymptotics of the logarithm of |Hom(G, H)| in terms of a simply expressed parameter of H.We also consider weighted versions of these results which may be viewed as statements about the partition functions of certain models of physical systems with hard constraints.

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 Z d admits multiple maximal-entropy Gibbs measures. This is a consequence of the following combinatorial result: if a proper 3-coloring is chosen uniformly from a box in Z d , conditioned on color 0 being given to all the vertices on the boundary of the box which are at an odd distance from a fixed vertex v in the box, then the probability that v gets color 0 is exponentially small in d.The proof proceeds through an analysis of a certain type of cutset separating v from the boundary of the box, and builds on techniques developed by Galvin and Kahn in their proof of phase transition in the hard-core model on Z d .Building further on these techniques, we study local Markov chains for sampling proper 3-colorings of the discrete torus Z d n . We show that there is a constant ρ ≈ 0.22 such that for all even n ≥ 4 and d sufficiently large, if M is a Markov chain on the set of proper 3-colorings of Z d n that updates the color of at most ρn d vertices at each step and whose stationary distribution is uniform, then the mixing time of M (the time taken for M to reach a distribution that is close to uniform, starting from an arbitrary coloring) is essentially exponential in n d−1 .

Write F for the set of homomorphisms from {0, 1} d to Z which send 0 to 0 (think of members of F as labellings of {0, 1} d in which adjacent strings get labels differing by exactly 1), and F i for those which take on exactly i values. We give asymptotic formulae for |F | and |F i |.In particular, we show that the probability that a uniformly chosen member f of F takes more than five values tends to 0 as d → ∞. This settles a conjecture of J. Kahn. Previously, Kahn had shown that there is a constant b such that f a.s. takes at most b values. This in turn verified a conjecture of I. Benjamini et al., that for each t > 0, f a.s. takes at most td values.Determining |F | is equivalent both to counting the number of rank functions on the Boolean lattice 2 [d] (functions f : 2 [d] −→ N satisfying f (∅) = 0 and f (A) ≤ f (A ∪ x) ≤ f (A) + 1 for all A ∈ 2 [d] and x ∈ [d]) and to counting the number of proper 3-colourings of the discrete cube (i.e., the number of homomorphisms from {0, 1} d to K 3 , the complete graph on 3 vertices).Our proof uses the main lemma from Kahn's proof of constant range, together with some combinatorial approximation techniques introduced by A. Sapozhenko.

Let I be an independent set drawn from the discrete d-dimensional hypercube Q d = {0, 1} d according to the hard-core distribution with parameter λ > 0 (that is, the distribution in which each independent set I is chosen with probability proportional to λ |I| ). We show a sharp transition around λ = 1 in the appearance of I: for λ > 1, min{|I ∩ E|, |I ∩ O|} = 0 asymptotically almost surely, where E and O are the bipartition classes of Q d , whereas for λ < 1, min{|I ∩ E|, |I ∩ O|} is asymptotically almost surely exponential in d. The transition occurs in an interval whose length is of order 1/d.A key step in the proof is an estimation of Z λ (Q d ), the sum over independent sets in Q d with each set I given weight λ |I| (a.k.a. the hard-core partition function). We obtain the asymptotics of Z λ (Q d ) for λ > √ 2 − 1, and nearly matching upper and lower bounds for λ √ 2 − 1, extending work of Korshunov and Sapozhenko. These bounds allow us to read off some very specific information about the structure of an independent set drawn according to the hard-core distribution.We also derive a long-range influence result. For all fixed λ > 0, if I is chosen from the independent sets of Q d according to the hard-core distribution with parameter λ, conditioned on a particular v ∈ E being in I, then the probability that another vertex w is in I is o(1) for w ∈ O but Ω(1) for w ∈ E.