25 problems

Let $G$ be a graph, and let $\mathbf{F}$ be a uniformly random forest chosen from all forests of $G$. For edges $e,f\in E(G)$, negative correlation conjecture. … This conjecture as…

The spanning-forest model is mapped perturbatively to an $OSP(1|2)$ sigma-model and, formally, to the $N$-vector model at $N=-1$. The usual correspondence fails nonperturbatively b…

Let $w_0({\cal L})$ denote the relevant critical-point parameter for a regular two-dimensional lattice ${\cal L}$ of coordination number $r$. The inverse-coordination conjecture. T…

Consider the spanning-forest model in two dimensions, with $L$ the strip width and $w_B(L)$ the relevant endpoint of its limiting zero curve. The asymptotic-freedom conjecture. The…

Let $f(w)$ be the spanning-forest free energy per site on a regular lattice of dimension $d\geq2$ and coordination number $r$, and let $f_0$ denote the entropy per site for spannin…

Uniform forest p-NC conjecture. The measure $\mathbb{P}_{\mathrm{UF}}$ satisfies the p-NC property.

Shteiner–Shteyner's conjecture. If $G=(V,E)$ is bipartite, then

Charikar–Liu–Liu–Vuong conjecture. A $1/\operatorname{poly}(n)$ fraction of the $k$-component forests of the grid graph have balanced component sizes.

Mohr–Pardey–Rautenbach conjecture. There exists a copy $F'$ of $F$ in $K_n$ such that

Let $m$ be the number of marked vertices in a graph. A quadratic spanning forest identity of the indicated type has coefficients as free variables, where the left-hand side consist…

Let $G$ be a $d$--regular graph on $n$ vertices, and let $F(G)$ denote its number of spanning forests. Bencs–Csikvári conjecture. … This is the proposed sharp analogue for spanning…

Let $K_n$ be a complete graph of order $n$, and let $c:E(K_n)\to \{-1,1\}$ be a zero-sum labeling satisfying … Let $F$ be a spanning forest of $K_n$ with maximum degree $\Delta$. M…

Let $K_n$ be a complete graph, let $c:E(K_n)\to\{-1,1\}$ be a zero-sum labeling satisfying … and let $F$ be a spanning forest of $K_n$. Write $\Delta(F)$ for the maximum degree of…

Let $\Gamma$ be the Temperley space-filling curve and let the Temperley CRSF be the corresponding Temperley cycle-rooted spanning forest. Let $\kappa'$ be the parameter from the in…

Genus-one twist-operator conjecture. In genus $1$, there might be a way to construct a model in which the twist operator is present, unlike the straightforward generalisation to hi…

Let $G$ be a general $\mathbb{Z}^2$-periodic graph, and consider the spanning-forest measure with slope parameter. A configuration has zero slope when its slope is $0$. Zero-slope…

Let $k\in[0,1)$, and consider the conductances assigned to edges and masses assigned to vertices by Equations … in the rooted spanning forest model. Complete-parametrization conjec…

Stronger negative-association conjecture. The measures $\operatorname{USF}$ and $\operatorname{UCS}$ are negatively associated in one or both of these further senses.

Spanning-forest critical-point conjecture. The point $(q,v)=(0,-3)$ is a feature of the thermodynamic-limit critical manifold; equivalently, the diced-lattice spanning-forest probl…

Let $\mathcal G$ be a graph. Say that $\mathcal G$ is $\Delta$-SOS if every relevant Rayleigh difference $\Delta G\{e,f\}$ has the required sum-of-squares representation, and say t…

Let $\mathcal G$ be a graph, and let $e$ and $f$ be distinct edges. Write $G=F(\mathcal G;\mathbf y)$ for the generating polynomial of its spanning forests, and let $\mathscr S$ be…

Let $n$ denote the number of components in a spanning forest, and let $\epsilon$ measure the distance from the critical coupling through $g=g_c e^{-\epsilon}$. For the cases analys…

Consider the spanning-forest model on random planar graphs, with coupling parameter $t$ and graph degree fixed arbitrarily. The preceding construction uses the graph's cyclic edge…

MNLP partition-function conjecture.

Let $G$ be a (weighted) graph. Let $\mathcal{F}$ be the set of all spanning forests of $G$, and let $\mathcal{B}$ be the probability distribution on spanning forests in which each…