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The strong lattice dependence of the Landau gauge propagator is likely due to the fact that this kind of strong deterministic gauge fixing produces long-range effects due entirely to the requirement of the gauge that $\partial_\mu A^\mu$ vanishes exactly, rather than statistically. This can produce artifactual correlations at long distances, due entirely to the gauge fixing. It should disappear in $\xi$ gauges with $\xi\ne 0$ (reappearing in the $\xi\rightarrow 0$ limit), since $\xi$ gauges make the divergence of $A$ white noise, and so prevent long-distance correlations at a $\xi$ dependent radius.
That these correlations are artifactual is obvious because they do not appear in gauge unfixed studies. The physical observables and the degree of lattice correlations between different regions are gauge invariant concepts, and they vanish when you do no gauge fixing.
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