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

TL;DR

This paper investigates the behavior of the chemical distance in random interlacements on high-dimensional lattices at low intensity, establishing bounds that support conjectures about its scaling limit as the intensity approaches zero.

Contribution

It provides the first rigorous bounds on the chemical distance in low-intensity random interlacements, confirming the conjectured scale and introducing a local lower bound technique.

Findings

Upper bound of order u^{-1/2} for the chemical distance

Lower bound of order u^{-1/2+ ext{small positive}}

Probabilistic bounds relevant for low-intensity geometric analysis

Abstract

In $\mathbb{Z}^d$ with $d\ge 5$, we consider the time constant $\rho_u$ associated to the chemical distance in random interlacements at low intensity $u \ll 1$. We prove an upper bound of order $u^{-1/2}$ and a lower bound of order $u^{-1/2+\varepsilon}$. The upper bound agrees with the conjectured scale in which $u^{1/2}\rho_u$ converges to a constant multiple of the Euclidean norm, as $u\to 0$. Along the proof, we obtain a local lower bound on the chemical distance between the boundaries of two concentric boxes, which might be of independent interest. For both upper and lower bounds, the paper employs probabilistic bounds holding as $u\to 0$; these bounds can be relevant in future studies of the low-intensity geometry.