5 problems

$p$-rotor recurrence conjecture. For almost every such $\rho$, the corresponding $p$-rotor walk on $\mathbb{Z}^2$ visits every vertex infinitely many times almost surely.

Quenched scaling conjecture. For almost every such $\rho$,

Let $\mathsf{G}$ be a finite graph on $\mathsf{N}$ vertices, let $D$ be its adjacency matrix, and let $\mathsf{T}_i$ be a periodic tree directed cover of $\mathsf{G}$ with root of…

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…