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Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models. (English) Zbl 1260.60186

Authors’ abstract: We study the diagonal heat-kernel decay for the four-dimensional nearest-neighbor random walk (on \(\mathbb{Z}^{4}\)) among i.i.d. random conductances that are positive, bounded from above but can have arbitrarily heavy tails at zero. It has been known that the quenched return probability \(\operatorname{P}_{\omega }^{2n}\left( 0,0\right) \) after \(2n\) steps is at most \(C\left( \omega \right) n^{-2}\log n\), but the best lower bound till now has been \(C\left( \omega \right) n^{-2}\). Here, we will show that the \(\log n\) term marks a real phenomenon by constructing an environment, for each sequence \(\lambda _{n}\rightarrow \infty ,\) such that \[ \operatorname{P}_{\omega }^{2n}\left( 0,0\right) \geq C\left( \omega \right) \log \left( n\right) n^{-2}/\lambda _{n}, \] with \(C\left( \omega \right) >0\;\)almost surely, along a deterministic subsequence of \(n\)’s. Notably, this holds simultaneously with a (non-degenerate) quenched invariance principle. As for the \(d\geq 5\) cases studied earlier, the source of the anomalous decay is a trapping phenomenon although the contribution is in this case collected from a whole range of spatial scales.

MSC:

60K37 Processes in random environments
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F17 Functional limit theorems; invariance principles
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