A factor of i.i.d with uniform marginals and infinite clusters spanned by equal labels

TL;DR

This paper constructs an example of a factor of i.i.d. vertex-labeling on a 3-regular tree with uniform marginals, resulting in some infinite clusters after deleting edges between vertices with different labels.

Contribution

It provides a novel example of an FIID labeling with uniform marginals that produces infinite clusters upon edge deletion, illustrating complex clustering behavior.

Findings

Existence of FIID labeling with uniform marginals on $ ext{T}_3$

Some clusters remain infinite after deleting edges between differently labeled vertices

Demonstrates complex clustering phenomena in random labelings

Abstract

We give an example of an $\mathsf{FIID}$ vertex-labeling of $\mathbb{T}_3$ whose marginals are uniform on $[0,1]$, and if we delete the edges between those vertices whose labels are different, then some of the remaining clusters are infinite.