6 problems

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

Invasion percolation scaling conjecture. For each $r\geq2$, $\vec t\in\mathbb{R}^{r-1}$ and $\vec k\in\mathbb{R}^{d(r-1)}$, there exist constants $A,V,v$ and $\delta\in(0,1)$ such…

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

Unoriented percolation scaling conjecture. There exist constants $A,V,v$ and $\delta\in(0,1)$ such that the scaling relation (rhoscal) holds for $\rho^{(\mathrm{pe})}$.

Let $\Sigma\subset \mathbb{R}^d$ be a compact set that is not a removable set for $\Delta$ in $L^2(\mathbb{R}^d)$. A self-adjoint extension of…

Let $S(X_t)=\overline{\{x:X(t,x)>0\}}$ be the closed support of $X_t$, let $\partial S(X_t)$ denote its boundary, and let $L_t$ be the local-time measure associated with the bounda…