As a computer scientist and "armchair" mathematician, I'm trying to replicate the images found here of Abelian sandpiles on an infinite square lattice, where the initial configuration is $n$ chips on a single vertex and all others empty. My best program to compute $n = 2^{30}$ will take an estimated 542 years, so I'm trying to understand this paper, pages 3 and 4, which seem relevant.
So far I've managed to figure out:
Because I only care about square lattices, $d = 2$.
The "odometer function" $v(x)$ is the number of times a cell x has been toppled.
1.2 states: the final number of grains on a cell $x$ is the initial number of grains plus, for each neighbor $y$, $v(y) - v(x)$. Therefore knowing the odometer function for each cell can yield the final sandpile.
The odometer function can be found using 1.3, and the final configuration follows immediately.
My goal is to find the odometer function, but 1.3 and the surrounding text leave me with these questions:
Where can one obtain the "standard estimates" for the 5-point Laplacian $\Delta^1$?
What does it mean to say "$\delta_0$ is the characteristic function of the set $\{0\} \subset \mathbb{Z}^2$"? Is this merely the indicator function that identifies $(0,0)$?
1.3 defines $v_n$ as the "the pointwise minimum of all functions $v : \mathbb{Z}^2 \rightarrow \mathbb{N}$" satisfying some criteria (which relates to the previous two bullets). There are an infinite number of functions $\mathbb{Z}^2 \rightarrow \mathbb{N}$, so it's not possible to try all of them. How can the number of candidate functions be limited to something computationally reasonable?