On the exact orders of critical value in Finitary Random Interlacements

TL;DR

This paper determines the precise asymptotic behavior of the critical intensity in Finitary Random Interlacements on integer lattices as the expected fiber length grows, revealing dimension-dependent scaling laws.

Contribution

It establishes the exact order of the critical intensity in Finitary Random Interlacements for different dimensions, a key step in understanding phase transitions in this model.

Findings

For d ≥ 5, critical intensity u*(T) ~ T^{-1}.

For d = 4, u*(T) ~ T^{-1} log T.

For d = 3, u*(T) ~ T^{-1/2].

Abstract

In this paper, we prove the exact orders of critical intensity $u_*(T)$ in Finitary Random Interlacements (FRI) in $\mathbb{Z}^d, \ d\ge 3$ with respect to the expected fiber length $T$. We show that as $T\to\infty$, $u_*(T)\sim T^{-1}, \ d\ge 5;$ $u_*(T)\sim T^{-1}\log T, \ d=4;$ $u_*(T)\sim T^{-1/2},\ d=3.$ Our estimates also give the order of magnitude at which the percolative phase transition with respect to $T$ takes place.