correlation in an almost independent set of random variables Suppose I have a set of $n$ binary random variables $X_1, \ldots, X_n$ that sit on a line, and assume that $\Pr(X_i=0)=\delta$ for all $i$. In addition, assume that any two subsets of variables that are separated by at least one site are completely independent. For example:
$P(X_1, X_2, X_4, X_7) = P(X_1,X_2)\cdot P(X_4, X_7) = P(X_1,X_2)\cdot P(X_4)\cdot P(X_7)$. 

In other words, whenever we look at a multi-variable dist', we can "break" it into blocks of contiguous variables.

Finally, assume that for any subset of $k&lt;n$ variables, $\{X_{i_1}, \ldots, X_{i_k}\}$, we have $\Pr(X_{i_1}=X_{i_2}=\ldots=X_{i_k}=0) &gt; 0$.

In that case, can I find a lower bound on $P_n\equiv \Pr(X_1=X_2=\ldots=X_n=0)$ ?

In particular, I would like to know, if there is a threshold $\delta_0$, such that for every $\delta&gt;\delta_0)$, it must be that also $P_n&gt;0$, and how this threshold depends on $n$. 

A simple use of Lovaz local lemma says that if $\delta&gt;7/8$ then indeed $P_n&gt;0$. But I think that it can be improved. For example, for $n=3$, a direct calculation shows that for $n=3$, it is impossible that $P_3=0$ unless $\delta &lt; (\sqrt{5}-1)/2 \simeq 0.61$.

Can you think of any method to attack this problem?
I think Yury's beautiful example actually answers my question. It shows that $\delta_0$ does not go to 0 as $n\to \infty$. One can probably also use a similar construction to show that the same holds for any $\delta&lt;1/2$.

Thanks Yury!