First passage percolation, local uniqueness for interlacements and capacity of random walk

TL;DR

This paper establishes sharp lower bounds for first passage percolation distances in random interlacements on various graphs, with applications to local uniqueness and capacity of random walks.

Contribution

It provides asymptotically sharp lower bounds for FPP distances in random interlacements on broad classes of graphs, extending previous results and applying to low-dimensional cases.

Findings

FPP distance is comparable to graph distance with high probability on z^d.

Sharp lower bounds hold in the near-critical phase for a wide class of graphs.

Applications include large deviation bounds for local uniqueness and capacity of random walks.

Abstract

The study of first passage percolation (FPP) for the random interlacements model has been initiated in arXiv:2112.12096, where it is shown that on $\mathbb{Z}^d$, $d\geq 3$, the FPP distance is comparable to the graph distance with high probability. In this article, we give an asymptotically sharp lower bound on this last probability, which additionally holds on a large class of transient graphs with polynomial volume growth and polynomial decay of the Green function. When considering the interlacement set in the low-intensity regime, the previous bound is in fact valid throughout the near-critical phase. In low dimension, we also present two applications of this FPP result: sharp large deviation bounds on local uniqueness of random interlacements, and on the capacity of a random walk in a ball.