18 problems

Let $G=K_n$ be the complete graph on $n$ vertices. Independently assign each edge weight $n^{1+gamma}$ with probability $1/n$ and weight $1$ otherwise. Let $mathbb T$ be the set of…

Aldous–Lyons conjecture. The UST of $(G,o)$ is one-ended if and only if $G$ is one-ended.

Let $K_m$ be the complete graph on $m$ vertices, let $\mathcal{T}$ be its uniform spanning tree, and let $d_{\mathcal{T}}(x,y)$ denote the distance between vertices $x$ and $y$ in…

Uniform spanning tree scaling conjecture. There exist constants $A,V,v$ and $\delta\in(0,1)$ such that (rhoscal) holds for $\rho^{(\mathrm{st})}$.

Let the UST Peano curve be the interface associated with a uniform spanning tree and its dual on a planar lattice, and let $\textrm{SLE}_8$ denote stochastic Loewner evolution with…

Let chordal $\textrm{SLE}_\kappa$ denote the random curve generated by chordal stochastic Loewner evolution with parameter $\kappa\geq 0$, and let the UST Peano curve be the interf…

Let $(W_{\vec e})_{\vec e\in\vec E}$ be i.i.d. weights with continuous distributions supported in $(0,\infty)$, rather than i.i.d. Exponential$(1)$ weights. Let the main theorem as…

For $kappa=8$, let $U(\lambda)$ be a solution of the relevant differential equation in the cross-ratio $lambda$, and interpret the constant solution as the partition function for t…

Let $G=K_n$ be the complete graph, with independent edge weights taking the value $n^{1+\beta}$ with probability $1/n$ and the value $1$ otherwise. Let $\beta$ vary with $n$, and l…

BCK2's conjecture. A similar combination of the various exponents should appear in sub-Gaussian annealed heat-kernel bounds for the random walk on the two-dimensional uniform spann…

Let $K_n$ be the complete graph on $n$ vertices, let $(U_e)_{e\in E(K_n)}$ be independent uniform random variables, and assign edge weights … with $0<\gamma<1/3$. Let the UST be th…

The spanning-tree and GFF scaling-limit conjecture. Under an appropriate scaling as $n\to\infty$, the 4-tuple $(M,P,\phi_1,\phi_2)$ converges in law to a unit boundary length…

Edge-Wilson speed-up conjecture. If Wilson's algorithm has this property on $G$, then the expected time taken by Edge-Wilson$(G)$ is asymptotically $25\%$ less than the expected ti…

Let $t_n$ be a uniformly sampled tree of size approximately $n$ in the square-torus uniform spanning-tree model, and let $w(t_n)$ be its Euclidean width. The UST width exponent con…

Condition on the event $|{\bf t}(r)|=n$, and let ${\sf Canonical}({\bf t}^{<n}(r))$ be the canonical embedding of the root component obtained by edge evaporation. The massive…

Let $({\bf T},r)$ be a rooted uniform spanning tree of a graph, and obtain ${\bf t}^{<n}(r)$ by successively removing outgoing edges at independent uniformly chosen non-root vertic…

Let $G=(V,E)$ be a connected graph on $n$ vertices, and let $f4a1$ denote a uniform spanning tree of $G$. Write $\tau_v$ for the hitting time of a vertex $v$, define the target tim…

Let $G=(V,E)$ be an infinite connected graph, and let $G_n$ be a finite exhaustion of $G$, meaning that each $G_n$ is finite, $G_n\subset G_{n+1}$, and $\bigcup_n G_n=G$. Let the U…