11 problems

Correlation-length conjecture. The correlation length grows like

Hyperbolic critical-density conjecture.

Let $Z$ be a Poisson point process of density $1$ in the plane, let its Voronoi tessellation be formed from the points of $Z$, and colour each Voronoi cell black independently with…

Benjamini–Kalai–Schramm conjecture. Knowing the Voronoi tessellation, but not the colouring of its cells, typically gives almost no information about whether the colouring contains…

Let $\alpha_4^{an}(r,n)$ denote the annealed probability of the relevant four-arm event from scale $r$ to scale $n$, for $1\leq r\leq n$. The four-arm exponent conjecture. One has…

Benjamini–Kalai–Schramm conjecture. With high probability, the quenched crossing probabilities are very close to the annealed crossing probabilities.

Kira–Neves–Schonmann's conjecture. If $\sigma$ and $\tau$ are constants, not necessarily equal, then for all possible such $\sigma$ and $\tau$,

Consider Voronoi percolation in a region $S$ with a point configuration whose points are independently perturbed according to Brownian motions run for time $t=t(n)$. Let the event…

Absolute-continuity invariance conjecture. Changing the uniform sampling measure to a uniformly absolutely continuous measure should not affect crossing probabilities in the limit…

Voronoi noise-sensitivity conjecture. Voronoi percolation is noise sensitive at criticality.

Let $\rho$ be the probability that each tile in the Poisson–Voronoi tessellation of ${\mathbb R}^d$ is coloured red, independently of the other tiles, and let $\rho_{\mathrm{c}}$ d…