On the internal distance in the interlacement set

Abstract

We prove a shape theorem for the internal (graph) distance on the interlacement set ${\mathcal{I}}^u$ of the random interlacement model on ${\mathbb{Z}}^d$, $d\ge 3$. 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.

26 pages, 3 figures

References in corpus (3)

• On the fragmentation of a torus by random walk
• Connectivity properties of random interlacement and intersection of random walks
• The effect of small quenched noise on connectivity properties of random interlacements

Cited by in corpus (16)

• Soft local times and decoupling of random interlacements
• On chemical distances and shape theorems in percolation models with long-range correlations
• Connectivity properties of random interlacement and intersection of random walks
• On scaling limits and Brownian interlacements
• The effect of small quenched noise on connectivity properties of random interlacements
• Biased random walk on the interlacement set
• On the range of a random walk in a torus and random interlacements
• Quenched large deviations for simple random walks on percolation models including long-range correlations
• On Chemical Distance and Local Uniqueness of a Sufficiently Supercritical Finitary Random Interlacement
• On (non-)monotonicity and phase diagram of finitary random interlacement
• Percolative properties of Brownian interlacements and its vacant set
• Percolation for the Finitary Random interlacements
• Euclidean and chemical distances in ellipses percolation
• All spatial random graphs with weak long-range effects have chemical distance comparable to Euclidean distance
• Some Rigorous Results on the Phase Transition of Finitary Random Interlacement
• Chemical distance in geometric random graphs with long edges and scale-free degree distribution