11 problems

Let the Abelian sandpile model on the square lattice $cmathbb{Z}^2$ have a recurrent sandpile configuration, and let its sandpile density mean the expected number of grains per ver…

Finite-time convergence conjecture. Then $u(t,\cdot)$ converges to $u_\phi$ in finite time.

Let $B_o(r)$ be a nonempty initial subset of the Sierpinski gasket, and independently place either $4-\delta$ or $4+\delta$ chips at each vertex with probability $1/2$. The resulti…

Fix $k\in[4,12)$, let $K$ be the limit fractal of the Sierpinski gasket, and let $\nu$ be its standard self-similar measure. Consider the rescaled sandpile configurations … These c…

Let a fixed state space support four Laplacian growth models: internal diffusion-limited aggregation (IDLA), rotor-router aggregation, divisible sandpiles, and abelian sandpiles. F…

A growth pattern is modeled on a lattice or continuous domain, and its growth is described by a gradient. A tropical function is a piecewise-linear function whose gradients encode…

Let $\vec{2}_{2n\times 2n}$ denote the all-$2$s configuration on the $2n\times 2n$ sandpile grid graph. Let $(a_n)_{n\geq 1}$ be the sequence of odd integers such that $2^n a_n^2$…

Let $bb_c$ be the supremum of densities for which a Poisson sandpile stabilizes, and let $b6_c(\mathbb{Z}^2)$ be the deterministic threshold density on the infinite square lattice.…

Let $b6_s$ denote the stationary density of the infinite-volume abelian sandpile on $b6^2$. A formula for $b6_s$ reduces its evaluation to an integral that has been numerically eva…

Let $b6_c$ be the threshold density of the fixed-energy sandpile on the square grid, and let $b6_s$ be the stationary density of the classical abelian sandpile on $b6^2$. Density c…

Dimensional reduction conjecture. There exists a constant $\lambda=\lambda_d<1$ such that for all $n\geq 1$ there exists $m\geq 1$ such that all but…