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We prove for the contact process on $Z^d$, and many other graphs, that theupper invariant measure dominates a homogeneous product measure with largedensity if the infection rate $\lambda$ is sufficiently large. As aconsequence, this measure percolates if the corresponding product measurepercolates. We raise the question of whether domination holds in the symmetriccase for all infinite graphs of bounded degree. We study some asymmetricexamples which we feel shed some light on this question. We next obtainnecessary and sufficient conditions for domination of a product measure for``downward'' FKG measures. As a consequence of this general result, we showthat the plus and minus states for the Ising model on $Z^d$ dominate the sameset of product measures. We show that this latter fact fails completely on thehomogenous 3-ary tree. We also provide a different distinction between $Z^d$and the homogenous 3-ary tree concerning stochastic domination and Isingmodels; while it is known that the plus states for different temperatures on$Z^d$ are never stochastically ordered, on the homogenous 3-ary tree, almostthe complete opposite is the case. Next, we show that on $Z^d$, the set ofproduct measures which the plus state for the Ising model dominates is strictlyincreasing in the temperature. Finally, we obtain a necessary and sufficientcondition for a finite number of variables, which are both FKG andexchangeable, to dominate a given product measure.Comment: 27 page

Stochastic domination: the contact process, Ising models and FKG measures☆