12 problems

Consider the arboreal gas $\mathcal{A}$ on a two-dimensional graph, whose configurations are forests and whose edge parameter is conventionally $x_e=\beta$ for every edge $e$. Arbo…

Let $G$ be a graph and let $a,b,c,d$ be vertices of $G$. For the arboreal gas measure $\mu^{\text{F}}_{G,\lambda}$, the notation $\mu^{\text{F}}_{G,\lambda}(a,b)$ and similar expre…

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…

For a finite graph $G$, let $\nu_{\beta,G}$ be the arboreal gas model with parameter $\beta$, defined by … For vertices in $G$, their connection probability is the probability that…

Large-$\beta$ tree-count conjecture. For $\beta$ very large, the number of infinite trees in the arboreal gas should be the same as the number of infinite trees in the uniform span…

High-dimensional arboreal-gas conjecture. In dimensions $d\geq 5$, the supercritical arboreal gas contains infinitely many infinite trees.

Let $Q_{\beta,a}$ be the probability measure used in the hyperbolic superspace representation, with parameter $a\geq 0$, partition function $Z_{\beta,a}$, and field variables $t_k$…

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…