.

In Section \[sec:adapt.mixing\], we will consider a more general case of a correlated, but strongly mixing random potential. Let $F_{V,x}(t) = \pr{ V(x;\omega) \leq t}$, $x\in\DZ^d$, be the marginal probability distribution functions (PDF) of the random field $V$, and $F_{V,x}(t\,|\, \fF_{\ne x}) := \pr{ V(x;\om) \le t \,|\, \fF_{\ne x}}$ the conditional distribution functions (CDF) of the random field $V$ given the sigma-algebra $\fF_{\ne x}$ generated by random variables $\{V(y;\om), y\ne x\}$. Our assumptions on correlated potentials are summarized as follows:

1.  The marginal CDFs are uniformly Hölder-continuous: for some $b>0$, \[eq:cond.continuity.V\] \_[x\^d]{} \_[a]{} \^b.

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1.  (Rosenblatt strong mixing condition.) For any $L\ge 1$ and pair of balls $\ball_{L}(u)$, $\ball_{L}(v)$ with $d(u,v)\ge L$, for any events $\CE_{u,L}\in\fF_{\ball_{L}(u)}$, $\CE_{v,L}\in\fF_{\ball_{L}(v)}$ \[eq:Cmix\] | - | C e\^[- \^2 L]{}.

This mixing condition can be relaxed to a power-law decay if only a polynomial decay of EF correlators is to be proven. (The polynomial decay of EF correlators is *not* the decay of eigenfunctions which we always prove to be exponential.)

Balls and boundary conditions {#ssec:balls}
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Unless otherwise specified, below we make use of the max-norm $\|x\| = \max_{1\leq j \leq d} |x_j|$ and the distance $\rd(\cdot,\cdot)$ induced by it. 