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Weakly random media affect propagation of waves and particles over long distances in a non-trivial way. When medium correlations decay rapidly the long time behavior is reasonably well understood: both waves and particles behave, loosely speaking, diffusively, and most non-trivial phenomena happen on the same time scale. Slow decay of media correlations seems to produce an interesting effect when single particle behavior becomes non-trivial on a much shorter time scale than the multi-particle behavior. Similarly, the wave phase becomes random long before wave energy randomizes. Moreover, while some quantities acquire fractional Brownian limits because of the slow correlations decay, others keep their diffusive limits. I will describe some rigorous as well as formal results in these directions. This is a joint work in progress with T.Komorowski.