7 problems

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 $(W_{\vec e})_{\vec e\in\vec E}$ 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…

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…