5 problems

Let $d_0,c>0$ be universal constants. For every integer $d\geq d_0$, let $G$ be a connected $n$-vertex graph with minimum degree at least $d$, and let $\mathcal{T}$ be a uniformly…

Let $\varepsilon>0$ be real. There is a constant $d_{\varepsilon}>0$ such that for every integer $d\geq d_{\varepsilon}$, every connected $n$-vertex $d$-regular graph $G$, and a un…

Let $K_n$ be the complete graph on $n$ vertices, let $\omega$ be the edge-disorder environment, and let ${{\mathcal T}}^\omega_{n,\beta_n}$ be the random spanning tree in random en…

Let the $m \times n$ grid graph be partitioned into $k$ connected pieces, and consider the spanning tree distribution on partitions, in which a partition has weight proportional to…

Let $m$ be the number of unbounded trees, and condition on $m\in\{2,3,4,5\}$. For cyclically labeled interfaces, define the sector widths … with the conventions…