8 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 $n$ be a natural number, let ${\mathcal T}$ be a set of trees on vertex set $[n]$, let $\mu$ be a probability distribution on ${\mathcal T}$, and let…

Let $n$ be a natural number, let ${\mathcal T}$ be a set of trees on vertex set $[n]$, let $\mu$ be a probability distribution on ${\mathcal T}$, and let…

Let $n$ be a natural number, let $d=d(n)$ satisfy $2\leq d\leq n-1$, let ${\mathcal T}$ be a set of trees of maximum degree $d$ on vertex set $[n]$, let $\mu$ be a probability dist…

Let $K_n$ be the complete graph, let $\mathsf{WST}$ be the weighted spanning tree model on an electric network with the random edge weights described in the surrounding text, and l…

Let $K_n$ be the complete graph on $n$ vertices, let $\beta_n=n^{\alpha}$ with $\alpha\in\left(1,\frac{4}{3}\right)$, and let $\mathsf{WST}_n^{\beta_n}$ denote the weighted spannin…

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…