On (non-)monotonicity and phase diagram of finitary random interlacement

TL;DR

This paper investigates the phase transition and monotonicity properties of Finitary Random Interlacement in dimensions 3 and 4, revealing non-monotonic behavior and providing numerical estimates for critical thresholds.

Contribution

It demonstrates non-monotonicity of FRI in certain dimensions and offers numerical evidence for a sharp phase transition and critical value estimates.

Findings

FRI is not stochastically monotone in dimensions 3 and 4

Numerical evidence supports a unique, sharp phase transition

Critical value inversely proportional to system intensity

Abstract

In this paper, we study the evolution of a Finitary Random Interlacement (FRI) with respect to the expected length of each fiber. In contrast to the previously proved phase transition between sufficiently large and small fiber length, we show that for $d=3,4$, FRI is NOT stochastically monotone as fiber length increasing. At the same time, numerical evidences still strongly support the existence of a unique and sharp phase transition on the existence of a unique infinite cluster, while the critical value for phase transition is estimated to be an inversely proportional function with respect to the system intensity.