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Decoupling inequalities and interlacement percolation on \(G\times \mathbb Z\). (English) Zbl 1277.60183
Summary: We study the percolative properties of random interlacements on \(G\times \mathbb Z\), where \(G\) is a weighted graph satisfying certain sub-Gaussian estimates attached to the parameters \(\alpha >1\) and \(2\leq \beta \leq \alpha +1\), describing the respective polynomial growths of the volume on \(G\) and of the time needed by the walk on \(G\) to move to a distance. We develop decoupling inequalities, which are a key tool in showing that the critical level \(u _{\ast }\) for the percolation of the vacant set of random interlacements is always finite in our set-up, and that it is positive when \(\alpha \geq 1+\beta /2\). We also obtain several stretched exponential controls both in the percolative and non-percolative phases of the model. Even in the case where \(G=\mathbb Z^{d }, d\geq 2\), several of these results are new.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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