Weakly correlated Bernoulli field

Question

Let $\Lambda\subset\mathbb{Z}^{d}$ ($\Lambda$ is finite). Let $\left\{ \eta_{x}\right\} _{x\in\Lambda}$ be a field of dependent Bernoulli random variables. I assume that their correlation decays fast i.e. $$\text{Cov}(\eta_{x},\eta_{y})\leq ce^{-C|x-y|}.$$

I need a bound from below for:

$$\mathbb{E}\exp\left(\sum_{x\in\Lambda}f_{x}\eta_{x}\right),$$ where $f_{x}\in\mathbb{R}$ (I can assume that these are bounded if needed). Ideally it would be some comparison with the i.i.d. case. For example: $$\mathbb{E}\exp\left(\sum_{x\in\Lambda}f_{x}\eta_{x}\right)\geq \prod_{x\in\Lambda}\mathbb{E}\exp\left(f_{x}\eta_{x}\right) - \text{"covariance term"}.$$

Or at least $$\mathbb{E}\exp\left(\sum_{x\in\Lambda}f_{x}\eta_{x}\right)\geq \mathbb{E}\exp\left(\sum_{x\in\Lambda_1}f_{x}\eta_{x}\right) \mathbb{E}\exp\left(\sum_{x\in\Lambda_2}f_{x}\eta_{x}\right)-\text{sth},$$ for some partition $\Lambda_1 \cup \Lambda_2 = \Lambda$.

Answer 1

Jensen's inequality gives a lower bound, but it might be too trivial for your needs. For comparing your expectation with the independent case, one can use the method of cluster expansions in statistical mechanics. Your model seems more like an Ising model than a field of Bernoulli variables.