14 problems

Let $\mathcal M_k$ be the supercritical random planar map, and let its free uniform spanning forest (FUSF) be the forest obtained from the free boundary condition. FUSF connectedne…

Let $G$ be a transitive transient graph. For $\varepsilon>0$, let an $\varepsilon$-$\operatorname{WUSF}$ be obtained by independently retaining each edge of a wired uniform spannin…

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…

Let $H$) be a finite connected graph and let $\mathbb{T}^d$ be the $d$-regular tree. For $w>0$, define the edge-weight function $\hat w$ on $G=\mathbb{T}^d\square H$ by assigning w…

Tang's lightness conjecture. The trees of $\mathsf{FUSF}$ should be non-light when $\mathsf{WUSF}\ne\mathsf{FUSF}$.

Tang's connectedness conjecture. For every $k\ge 3$ and every finite vertex-transitive graph $H$, the $\mathsf{FUSF}$ on $\mathbb{T}^k\times H$ is connected almost surely.

Let $\mathbb{T}_{b+1}$ be the $(b+1)$-regular tree and let $H$ be a finite connected graph. Write $\mathbb{T}_{b+1}\square H$ for their Cartesian product, and let…

FUSF heaviness and branching conjecture. Each tree in the $\operatorname{FUSF}$ is $\Gamma$-heavy and has branching number bigger than one almost surely.

Let $G$ be a locally finite, connected graph. Write $\operatorname{\mathsf{WSF}}(G)$ for the wired uniform spanning forest on $G$, let $\mathfrak{F}_{\mathrm w}(G)$ denote a sample…

Let $\mathbb G_1$ and $\mathbb G_2$ be $d$-dimensional transitive graphs, let $r_1,r_2\geq 1$, and let $\mathfrak F_1$ and $\mathfrak F_2$ be the uniform spanning forests of…

Let $\mathbb G$ be a $d$-dimensional transitive graph for some $d>8$, let $\mathfrak F$ be its uniform spanning forest, and let $r\geq 1$. Write $\mathcal C_r(\mathfrak F)$ for the…

Let $T_{0}$ be the component of the origin in the uniform spanning forest on $\mathbb{Z}^{d}$, and let $\xi$ be the unique infinite ray from $0$ in $T_{0}$. The Benjamini–Lyons–Per…

Update-stable zero-one conjecture. For every update-stable Borel set $\mathscr A\subseteq \{0,1\}^E$, the probability that $\mathfrak F$ belongs to $\mathscr A$ is either zero or o…

Let $G$ be a unimodular transitive graph. The wired and free uniform spanning forest indistinguishability conjecture. The components of both the wired uniform spanning forest (WUSF…