Invariant monotone coupling need not exist

TL;DR

The paper demonstrates through an example that invariant monotone couplings between invariant random subgraphs of a Cayley graph may not exist, highlighting limitations in invariant coupling constructions.

Contribution

It provides the first known example showing the non-existence of invariant monotone couplings between certain invariant random subgraphs.

Findings

Invariant monotone couplings do not always exist.

An explicit Cayley graph example is constructed.

Invariant couplings can fail even when monotone couplings exist.

Abstract

We show by example that there is a Cayley graph, having two invariant random subgraphs X and Y, such that there exists a monotone coupling between them in the sense that $X\subset Y$, although no such coupling can be invariant. Here, "invariant" means that the distribution is invariant under group multiplications.