On monotonicity and couplings of random currents and the loop-$\mathrm{O}(1)$-model

TL;DR

This paper investigates the monotonicity properties of percolation models linked to the Ising model and introduces a new coupling between the double random current model and the loop-$ ext{O}(1)$-model, providing counterexamples and novel connections.

Contribution

It provides counterexamples to monotonicity in percolation models and establishes a new coupling between the double random current and loop-$ ext{O}(1)$-models.

Findings

Counterexamples to monotonicity properties in related percolation models

A new coupling between the double random current model and the loop-$ ext{O}(1)$-model

Insights into graphical representations of the Ising model

Abstract

Using recent couplings we provide counterexamples to monotonicity properties of percolation models related to graphical representations of the Ising model. We further prove a new coupling of the double random current model to the loop-$\mathrm{O}(1)$-model.