We prove a shape theorem for the internal (graph) distance on the?interlacement set $mathcal{I}^u$ of the random interlacement model on?$mathbb Z^d$, $dge 3$. We provide large deviation estimates for the?internal distance of distant points in this set, and use these?estimates to study the internal distance on the range of a simple?random walk on a discrete torus.
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机译：我们证明了关于在$ mathbb Z ^ d $，$ d ge 3 $上的随机交错模型的交错集$ mathcal {I} ^ u $的内部（图形）距离的形状定理。我们提供了该集合中远点内部距离的大偏差估计，并使用这些估计来研究离散圆环上简单随机游动范围内的内部距离。
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