10 problems

Consider critical percolation on a high-dimensional torus of side length $r$, let ${\cal C}_{\rm max}$ be a largest critical cluster, and choose a point uniformly from…

At criticality, let $C(0)$ be the cluster containing the origin, condition on $|C(0)|=n$, and scale space by $D^{-1}n^{-1/4}$, where $D$ is the model-dependent scaling constant. Le…

Let the incipient infinite cluster be obtained as the scaling limit of increasingly large finite critical percolation clusters, with lattice spacing scaled by $n^{-1/4}$ in dimensi…

Two-set p-capacity factorisation conjecture.

IIC hitting conjecture.

IIC scaling-limit conjecture. The IIC is super-Brownian or super-$\alpha$-stable motion conditioned to survive forever.

Let $X_{r,r}$ denote the portion of the incipient infinite cluster reaching distance $r$ through the box of radius $r$, and let … Here $bP_{\mathrm{IIC}}$ denotes probability under…

Let $\mathsf{FETIC}$ be the law of the configuration at the first exceptional time, and let $\mathsf{IIC}$ denote the incipient infinite cluster measure. Singularity conjecture. Th…

Exit-time scaling conjecture. The time spent on the other side of long edges is sufficiently short that these excursions do not affect the exit time in the scaling limit, in the st…

Let $\a0\mathbb{P}_{\mathsf{IIC}}$ and $\a0\mathbb{Q}_{p_c}$ be the measures defined, respectively, by conditioning on connection to a distant vertex and by the susceptibility-meas…