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Arboreal Gas is a type of (unrooted) random forest on a graph, where theprobability is determined by a parameter $\beta>0$ per edge. This model isessentially equivalent to acyclic Bernoulli bond percolation with a parameter$p=\beta/(1+\beta)$. Additionally, Arboreal Gas can be considered as the limitof the $q$-states random cluster model with $p=\beta q$ as $q\to 0$. A naturalquestion arises regarding the existence and performance of the weak limit ofArboreal Gas as the graph size goes to infinity. The answer to this questionrelies on the negative correlation of Arboreal Gas, which is still an openproblem. This paper primarily focuses on the negative correlation of ArborealGas and provides some results for specific graphs.

On Negative Correlation of Arboreal Gas on Some Graphs