7 problems

Consider minimal spanning forests on ${\mathbb Z}^d$, and let the number of infinite clusters refer to the clusters of the relevant minimal spanning forest. Newman–Stein conjecture…

Minimal spanning forest dimension conjecture. Above dimension $8$, the stochastic dimension of the resulting minimal spanning forest structure equals $8$.

Let $Q$ be the Poisson point set in Euclidean space and let $R_\infty$ be the associated minimal spanning forest, whose edges are formed from the limiting minimum spanning construc…

Let a Poisson point process (PPP) be given, and consider its minimal spanning forest (MSF), the infinite-volume analogue of the minimal spanning tree for random points. An Aldous–S…

Benjamini–Schramm FMSF conjecture. The graph $Gamma$ has uniqueness at $p_u$ if and only if its FMSF is connected. Moreover, this can occur only if

BLPS conjecture. Almost surely, every tree in $\mathsf{FSF}_G$ has infinitely many ends.

Let $\mathcal G=(V,E)$ be a transitive graph, and let $\Gamma\subset\operatorname{Aut}(\mathcal G)$ act transitively on $V$ and freely on $E$ with amenable edge stabilizers. Let…