14 problems

Let $G=(V,E)$ be a graph, let $boldsymbol{x}=xe:e otin E$ be positive edgeweights, and let the random cluster model on $G$ have parameter $q$. For distinct edges $e,f otin E$, writ…

Let a matroid belong to one of the listed classes: planar-graph matroids, graph matroids, regular matroids, sixth-root-of-unity matroids, half-plane-property matroids, or B-Rayleig…

Let $\mathsf G=(V,E)$ be a finite connected graph. For each independent edge set $S$ of $\mathsf G$, let $\omega_G(S)$ be the product of the sizes of the connected components of…

Let $\mathsf G$ be a graph and let $e,f$ be edges. Consider the Rayleigh differences $\Delta Z(\mathbf I\mathsf G)\{e,f\}$ and $\Delta Z(\mathbf B\mathsf G)\{e,f\}$ for the indepen…

Let $\mathsf G$ be a graph and let $e,f$ be edges of $\mathsf G$. Write $\Delta Z(\mathbf I\mathsf G)\{e,f\}$ for the Rayleigh difference of the independence-set partition function…

Let $\mathsf K_n$ be the complete graph and let $Z(\mathbf I\mathsf K_n;\mathbf y)$ be the partition function of its independent-set system. For edges $e,f$, let $\Delta Z\{e,f\}$…

Let $Z(\omega;\mathbf y)$ be Rayleigh, let $\widetilde Z$ be its symmetrization, and let $z(e)$ and $z(ef)$ be the auxiliary quantities defined in the source. Pairwise symmetrizati…

Let $Z(\omega;\mathbf y)$ be Rayleigh, and let $\widetilde Z$ be its symmetrization. For each $e\in E$ and each pair $\{e,f\}\subseteq E$, use the auxiliary quantities $z(e)$ and…

Let $Z(\omega;\mathbf y)$ be a partition function on a finite ground set $E$, and let $\mathsf A_1,\mathsf A_2$ be increasing events with disjoint support. Negative association con…

Symmetrization conjecture. If $Z(\omega;\mathbf y)$ is Rayleigh, then $\widetilde Z(\omega;\mathbf y)$ is Rayleigh. This is stated as an equivalent form of the Big Conjecture and r…

Let $E$ be a finite set of size $m$, let $\mathsf{B}(E)$ be its Boolean algebra, and let $\omega:\mathsf{B}(E)\to[0,\infty)$ be a nonnegative weight function. Define … The Rayleigh…

Let $X$ be an isotropic unconditional log-concave random vector in $\mathbb{R}^n$, meaning that its distribution is invariant under coordinate reflections and it has a logarithmica…

Positivity conjecture for $\mathcal{M}_{ef}(q)$. For $0\leq q\leq1$ and positive edgeweights $x_e$, one has

Seymour–Welsh conjecture. The correlation constant of every matroid is at most one: