9 problems

Consider stochastic sandpiles with uniform topplings on $\mathbb{Z}$. Let $\delta_0$ denote the unit mass at the origin, and let $D_n$ be the set of vertices toppled during stabili…

Let $d\geq 1$, start with $n$ particles at the origin of $\mathbb{Z}^d$, and stabilize them in $\mathbb{Z}^d$. Denote by $A_n$ the random set of visited sites, and let $\zeta_c$ be…

Shape theorem conjecture. There exists a compact subset $\widehat S\subset\mathbb{R}^d_+$ such that

Shape-coincidence conjecture. Coexistence is possible if and only if the one-type shapes coincide:

Let ORRW and OERW denote the two growth models discussed in the source, with evolution indexed by time. Asymptotic shape conjecture. For both ORRW and OERW, the evolution should le…

Boundary-range shape theorem conjecture. A shape theorem takes place on the scale

Asymptotic shape conjecture. For every $p>\overrightarrow{p_c}^{\text{alt}}(d+1)$, every $q<\overrightarrow{p_c}^{\text{alt}}(d+1)$, and every $\alpha\in(0,\alpha_c(p,q))$, there e…

Shape theorem conjecture. On the event $\{\zeta_t\text{ survives}\}$, almost surely, $t^{-1}(D_t\cap K_t)$ has an asymptotic shape.

Sufficiency conjecture. If, for all bounded $A,B$,