6 problems

Let $\kappa\in(0,8)$, and consider a multiple-SLE$_\kappa$ process growing $2N$ curves in the upper half-plane from points $x_1<x_2<\ldots<x_{2N}$, with SLE$_\kappa$ partition func…

Let $G=(V,E)$ be a regular hexagon on the triangular lattice with side lengths $\ell$, and let $\mathcal U$ be the event that there exists a vertex connected to each of the three a…

Let $G=(V,E)$ be an $m$-rhombus on the triangular lattice, let $f:V\to\{a,b,c,d\}$ be a uniform random coloring, and let $\mathcal U=\{12\leftrightarrow34\}$ and…

Let $G=(V,E)$ be an $n\times(n+1)$ rectangle, let $f:E\to\{a,b,c,d\}$ be a uniform random coloring, and let $\mathcal U=\{12\leftrightarrow34\}$ denote connectivity between opposit…

Ising crossing-probability conjecture. The crossing probabilities converge according to

Universal crossing conjecture. There is a universal constant $c>0$ such that the probability of a black left-to-right crossing is greater than $c$.