6 problems

Consider the rotor-router growth model on $\mathbb{Z}^d$, and let $A$ be its occupied region after particles are added at the origin. Rotor-router Euclidean-ball conjecture. The li…

Consider the deterministic rotor-router growth model on $\mathbb{Z}^d$, and compare its asymptotic occupied regions with those of internal diffusion-limited aggregation (IDLA). Rot…

Kordzakhia's limiting-shape conjecture. The influence region has a definite limiting shape $S\subset {{\mathbb R}}^2$ in the sense that, almost surely, $R_t$ converges to the shape…

Polygonal limiting-shape conjecture. For every

Robust-regime conjecture. Theorem $$ holds for all

Limiting-picture conjecture. There is a sequence of integers $a_n$ such that $a_n\uparrow\infty$ and $a_n/n\downarrow0$, and there is a function…