5 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…

Height-three percolation conjecture. There exists $p_c\in(0,1)$ such that, for all $p<p_c$,

Let $G=(V\cup\{s\},E)$ be a finite graph with designated sink vertex $s$, and let $\rho_G(p)$ denote the stationary average density for the $p$-toppling rule, where $p\in[0,1]$ and…

Let $G=(V\cup\{s\},E)$ be a finite graph with designated sink vertex $s$. Under the $p$-toppling rule, each edge incident to a toppled vertex independently sends a particle with pr…

In driven-dissipative dynamics, particles are repeatedly added and the configuration is stabilized on a finite graph with a designated sink, while fixed-energy dynamics starts from…