9 problems

Let $\zeta_{\tau}(h)$ denote the threshold density of the fixed-energy sandpile with constant initial condition $h$, and let $\zeta_s$ denote the stationary density of Dhar's abeli…

Let $\nu$ be the infinite-volume sandpile measure and let $\mathrm{rad}(\mathrm{Av}_o)$ be the radius of the avalanche set generated by adding a particle at the origin. Avalanche-r…

Let $\nu$ be the infinite-volume sandpile measure. For an avalanche started by adding a particle at the origin, let $S_o=\sum_{x\in\mathbb{Z}^d}n(o,x;\cdot)$ be the total number of…

Let $\nu$ be the infinite-volume sandpile measure on $\{0,1,\ldots,2d-1\}^{\mathbb{Z}^d}$, let $\mathrm{Av}_o=\{x\in\mathbb{Z}^d:n(o,x;\cdot)>0\}$ be the avalanche set generated by…

Logarithmic tail conjecture. For a general parameter $D$ and all $N$, there exists a column $n$ in $O(\log N)$ such that

Consider the Manna model on $\mathbb Z$. Let $\rho_s$ be its stationary density, let $\rho_{tr}$ be the transition density identified by the change in the long-time toppling behavi…

Polygonal limiting-shape conjecture. For every

Robust-regime conjecture. Theorem $$ holds for all

Critical-threshold conjecture. There exists a critical value $h_c=h_c(d)$ such that, for all $h<h_c$, the model is robust, while for all $h\geq h_c$, the model is explosive. Moreov…