20 problems

Let Zhang's sandpile model have random additions uniformly distributed on an interval $[a,b]$, with toppling threshold $E_c$, and let the system volume tend to infinity. Zhang's co…

In sandpile models, grains are added only when the dynamics eventually stops, as in the original Bak–Tang–Wiesenfeld model, and dissipation is localized at the boundaries. Vanishin…

Self-organized criticality density conjecture. The typical density $\rho_s$ in the steady state of the driven-dissipative system corresponds to $\rho_c$.

Let $c_{ij}$ be the coefficients associated with the four boundary types $(i,j)\in\{0,1\}^2$, and let $\overline{c_{ij}}$ denote their averages in the experiments. Mean identity. T…

The Zhang model has continuous non-negative height variables $z_i$, a common threshold height $z_i^c$, and random additions chosen from an interval $[a,b)$ with $0\leq a<b$. A site…

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…

In the two-dimensional sandpile model on the discrete upper half-plane, let $p^*(i;m)$ be the probability of height $i$ at $(0,m)$ under either the open or closed boundary conditio…

In the two-dimensional sandpile model, let $p(i)=\nu[\eta(o)=i]$ be the probability that the height at the origin is $i$, and let $I$ denote the unknown integral appearing in the f…

The critical dimension $d_c$ of a lattice model is the threshold above which its critical exponents are expected to take dimension-independent mean-field values. Sandpile critical-…

Mean-field avalanche-radius conjecture. When $d>4$, $\alpha=2$.

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…

Let $\rho_s$ denote the stationary density of the Manna model on $\mathbb Z$. Sadhu's conjecture. The stationary density satisfies … Sadhu et al. estimate $\rho_s=0.953\ldots$ by e…

Let $P_i$ denote the stationary probability that a site of the square-lattice Abelian sandpile model has height $i$, and let $\langle h\rangle$ denote the stationary mean height (p…

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…