8 problems

Convergence conjecture.

Let $T_r$ be the infinite $r$-regular tree, and let $\chi_{fiid}(T_r)$ denote its factor-of-iid chromatic number, the least number of colors in a proper coloring of $T_r$ produced…

Folklore uniqueness conjecture. This Gibbs measure is unique if and only if $w\geq w_c$; if $k=d+1$, the inequality should be interpreted as strict.

Let ${\mathbb Z}^\nu$ be the $\nu$-dimensional cubic lattice, and let $d$ denote the number of descendants in the regular-tree model studied in the paper. The authors obtain an exp…

Let $\mathbb{T}_{d}$ be the homogeneous $d$-ary tree with $d\geq 3$, and let ${\mathbf a}$ be an $L$-periodic rotor sequence. The regular-tree uniform-shift transience conjecture.…

Let $\mathbb{T}_{2}$ be the binary tree, and let ${\mathbf a}$ be an $L$-periodic rotor sequence. Define $\mathcal{A}_{\mathrm{rec}}^{\mathrm{conj}}$ to consist of all shifts of se…

Let $\mathcal{X}$ be the configuration space of the Williams–Bjerknes model on the regular tree $\mathbb{T}$, and let $\delta_emptyset$ and $\delta_\mathbb{T}$ denote the point mas…

Let $\lambda_l(\mathbb{T})$ and $\lambda_c(\mathbb{T})$ denote the local and global survival thresholds for the Williams–Bjerknes model on the regular tree $\mathbb{T}^d$. For…