7 problems

Let $G=(V,E)$ be a graph and let $u,v,w\in V$. Let $\mu^{\text{F}}_{G,\lambda}$ denote the arboreal gas measure with parameter $\lambda$, and write…

Let $G$ be a graph, let $T\subset V(G)$ be a set of posts, and let $\nu^{\text{F}}_{G,T}$ denote the alternate arboreal gas measure on red-blue edge-colourings, restricted so that…

Let $G$ be a graph, let $\widetilde{G}$ be its associated bunkbed graph, and let $\mu^{\text{F}}_{\widetilde{G},\lambda}$ denote the arboreal gas measure with parameter $\lambda$.…

Let $\beta>0$, and for distinct edges write $\mathbb{P}_{\beta}[e_1,\ldots,e_k]$ for the probability that all listed edges occur in the forest. Negative association conjecture. For…

Let the arboreal gas be defined on $\mathbb{Z}^2$, let $\beta>0$, and write $\mathbb{P}_{\beta}[i\leftrightarrow j]$ for the probability that $i$ and $j$ lie in the same tree. Let…

Let the arboreal gas be defined on $\mathbb{Z}^d$ with $d\geq 3$, and write $\mathbb{P}_{\beta}[i\leftrightarrow j]$ for the probability that $i$ and $j$ lie in the same tree. Gold…

Let the arboreal gas be defined on $\mathbb{Z}^d$ with $d\geq 3$, let $B_r$ be the radius-$r$ ball, and let $T_0$ denote the tree containing the origin. Write…