| The first problem we consider is from statistical physics. Write for the set of independent sets of the graph . For finite and > 0, the hard-core measure with activity on is given by mI=l I/Z ∀I∈I, where Z = is the appropriate normalizing constant. We say that this measure is hc().; For infinite a measure on is hc() if for I chosen from according to and for all finite W ⊆ V = V(G), the conditional distribution of I ∩ W given I ∩ (V{bsol}W) is (-a.s.) hc() on the independent sets of {lcub}w ∈ W : w I ∩ (V{bsol}W){rcub}. There is always at least one such measure. If there is more than one, the model is said to have a phase transition.; Dobrushin [9] (and later, independently, Louth [21]) showed that there is a phase transition in the hard-core model on (the usual nearest neighbor graph on) Zd for sufficiently large values of (depending on d). In other words, they showed that ld : =sup thehard-c oremodelwitha ctivitylon l: Zd doesnothave aphasetransition <∞. ; Up to now, all known bounds for (d) increased rapidly with d. However, it has been widely conjectured that (d) → 0 as d → . This is what we prove.; The second problem we consider comes from discrete probability. It was introduced by Benjamini, Häggström and Mossel [2] (and, in a different context, by Athanasiadis [1]).; Write for the set of homomorphisms from the d-dimensional Hamming cube {lcub}0,1{rcub}d to (the Hamming graph on) Z which send 0&barbelow; (the all-zero string) to 0 and for those which take on five or fewer values. (A homomorphism between graphs is an adjacency preserving map between vertex sets.) We show that |
