## Vacant set of random interlacements and percolation.(English)Zbl 1202.60160

The paper deals with the model of random interlacements consisting of a countable collection of doubly infinite trajectories on a multidimensional integer lattice. A nonnegative parameter $$u$$ measures how many trajectories enter the picture. The union of the supports of these trajectories defines the interlacement at level $$u$$. It is an infinite connected translation invariant random subset of the lattice. The author introduces a critical value $$u^{\ast}$$ such that the vacant set percolates for $$u$$ less then $$u^{\ast}$$ and does not percolate for $$u$$ greater then $$u^{\ast}$$. The main results of the paper show that $$u^{\ast}$$ is finite for three-dimensional lattice and strictly positive when the lattice dimension is greater than 6.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G50 Sums of independent random variables; random walks 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics

### Keywords:

Percolation; Random interlacement; Poisson point process
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### References:

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