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 $k<n$ variables, $\{X_{i_1}, \ldots, X_{i_k}\}$, we have $\Pr(X_{i_1}=X_{i_2}=\ldots=X_{i_k}=0) > 0$.
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 $\delta>\delta_0)$, it must be that also $P_n>0$, and how this threshold depends on $n$.
A simple use of Lovaz local lemma says that if $\delta>7/8$ then indeed $P_n>0$. 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 $\delta < (\sqrt{5}-1)/2 \simeq 0.61$.
Can you think of any method to attack this problem?
