Some Rigorous Results on the Phase Transition of Finitary Random Interlacement

TL;DR

This paper provides rigorous mathematical results on the phase transition behavior of Finitary Random Interlacement, identifying critical parameters and phase existence in different intensity regimes.

Contribution

It establishes the existence of a critical fiber length at high intensity and proves a non-trivial phase transition at low intensity for Finitary Random Interlacement.

Findings

Existence of a critical fiber length at high intensity

Asymptotic behavior of the critical fiber length as intensity increases

Global existence of a non-trivial phase transition at low intensity

Abstract

In this paper, we show several rigorous results on the phase transition of Finitary Random Interlacement (FRI). For the high intensity regime, we show the existence of a critical fiber length, and give the exact asymptotic of it as intensity goes to infinity. At the same time, our result for the low intensity regime proves the global existence of a non-trivial phase transition with respect to the system intensity.