On negative correlation of Arboreal Gas for specific parameters

Arboreal Gas is a type of (unrooted) random forest on a graph, where the probability is determined by a parameter $\beta >0$ per edge. This model can be considered as the limit of the $q$-states random cluster model with $p=\beta q$ as $q\to 0$. A natural question arises regarding the existence and performance of the weak limit of Arboreal Gas as the graph size goes to infinity. The answer to this question relies on the negative correlation of Arboreal Gas, which is still an open problem. This paper primarily focuses on the negative correlation of Arboreal Gas and provides some results for specific parameters.