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The chemical distance in random interlacements in the low-intensity regime

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Random interlacements is a Poissonian soup of doubly-infinite random walk trajectories on Zd, with a parameter u > 0 controlling the intensity of the Poisson point process. In a natural way, the model defines a percolation on the edges of Zd with long-range correlations. We consider the time constant associated to the chemical distance in random interlacements at low intensity u > 0. It is conjectured that the time constant times u^{1/2} converges to the Euclidean norm, as u ↓ 0. In dimensions d ≥ 5, we prove a sharp upper bound and an almost sharp lower bound for the time constant as the intensity decays to zero. Joint work with Eviatar Procaccia and Ron Rosenthal.

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