4 We consider families of {0, 1}-valued random variables indexed by
the vertices of countable graphs with bounded degree. First we show that if
these random variables satisfy the property that conditioned on what happens
outside of the neighborhood of each given site, the probability of seeing a 1
at this site is at least a value $p$ which is large enough, then this random
field dominates a product measure with positive density. Moreover the density
of this dominated product measure can be made arbitrarily close to 1, provided
that $p$ is close enough to 1. Next we address the issue of obtaining the
critical value of $p$, defined as the threshold above which the domination by
positive-density product measures is assured. For the graphs which have as
vertices the integers and edges connecting vertices which are separated by no
more than $k$ units, this critical value is shown to be $1 - k^k /(k +
1)^{k+1}$, and a discontinuous transition is shown to occur. Similar critical
values of $p$ are found for other classes of probability measures on ${0,
1}^{\mathbb{Z}}$. For the class of $k$-dependent measures the critical value is
again $1 - k^k /(k + 1)^{k+1}$, with a discontinuous transition. For the class
of two-block factors the critical value is shown to be 1/2 and a continuous
transition is shown to take place in this case. Thus both the critical value
and the nature of the transition are different in the two-block factor and
1-dependent cases.