Sandpile Projects Ideas

Acyclic orientations

The paper G-parking functions, acyclic orientations and spanning trees gives a bijection between maximal G-parking functions and acyclic orientations of a graph. A maximal G-parking function is the same as a maximal superstable configuration or a minimal recurrent configuration or a minimal alive configuration.

Give an exposition of the main result of this paper in terms most closely related to our class.
Does their idea extend to all recurrents?
Can the result be modified to account for the minimal recurrents of a directed graph?

Identity Element

Experimentally, one sees that the identity configuration for a rectangular grid graph has a large rectangle inside it, however no one has been able to prove this. Here are some computer experiments that would be interesting.

How does this result depend on the precise border of the rectangle? What if the border were approximately a rectangle? Would the rectangle inside the identity element disappear?
Does the result hold in three dimensions (and higher)?

Embedded components

Let $\ell_1,\dots,\ell_n$ be the columns of the reduced Laplacian matrix for a sandpile graph $(\Gamma,s)$ . Define the ideal $J=\langle x^{\ell_i^+}-x^{\ell_i^-}:i=1,\dots,n\rangle$ . From the theory of lattice ideals, we know that the affine toppling ideal, $I$ , for $\Gamma$ is the saturation of $J$ by $p=\prod_{v\in\widetilde{V}}x_v$ . We also know that $I=J+\langle x^{\mathbf{b}}-1\rangle$ for any burning configuration $\mathbf{b}$ (we will fix the minimal burning configuration).

Characterize the minimal $k$ such that $p^k(x^{\mathbf{b}}-1)\in I$ .
Let $P$ be the embedded component of the primary decomposition of $I$ , and let $M$ be the ideal generated by the indeterminates. Characterize the minimal $k$ for which $M^k\subseteq P$ . (One could ask the same question in the homogeneous case, in which one starts with binomial generators corresponding to the columns of the full Laplacian.

The Bombay Trick

Consider the sandpile model on a grid. Dhar discovered a way method—called the Bombay trick—by which to count the proportion of recurrent elements that have no sand at a given vertex. The method is described in Height correlations in the Abelian sandpile model and in section 3.7 of Mathematical aspects of the abelian sandpile model. Here is related paper (see page 5).

Give an exposition of this result.
Try applying the trick to a different class of graphs.

Riemann-Roch

Investigate how the Riemann-Roch theorem for graphs breaks down for directed graphs. This project would be an expansion of problem 2 from HW 7.

Lattice points in polytopes

To find the complete linear system of a divisor $D$ , one must solve a system of linear inequalities over the integers: $|D|= \{E: E = D+\Delta\sigma, E\geq 0\}$ . Geometrically, we are translating the Laplacian lattice by $D$ , then intersecting with the positive orthant. The elements of the linear system are then just the lattice points inside of a simplex (I think). One could also consider the collection of scripts $\{\sigma: \Delta\sigma\geq -D\}$ and consider this set of lattice points.

Compute some examples. Draw some pictures for low-dimensional examples.
What does Riemann-Roch say in this context?
What do the formulas for counting lattice points in polytopes say?

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