6 problems

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…

Uniqueness conjecture. The $1$-dependent $4$-coloring is unique.

Let $d\geq 3$, and consider uniformly sampled homomorphism height functions on a domain in $\mathbb{Z}^d$ with zero boundary conditions. Bounded-fluctuation conjecture. The fluctua…

Let $h_L$ be a uniformly sampled homomorphism height function on the two-dimensional domain $\Lambda(L)$ with zero boundary conditions, and let its level lines be the interfaces be…