Dhar's Burning test is a bijection between the spanning trees of a certain graph and the recurrent states Abelian sandpile model. I would like some help working out this bijection in different cases:
Square graph
One we pick a root there are 4 spanning trees:
o-o o-o o-o o o o-o
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x-o x-o x-o x-p x-o
Also the recurrent states should be $\mathbb{Z}^3/V\mathbb{Z}^3$ where $V$ is the lattice $\mathbb{Z}(-2,1,0) + \mathbb{Z}(1,-2,1) + \mathbb{Z}(0,1,-2)$. There should be 4 such states since:
$$ \left| \begin{array}{ccc} -2 & 1 & 0 \\ 1 & -2 & 1 \\ 0 & 1 & -2 \end{array}\right| = 4$$
How do I write down explicit representatives of $\mathbb{Z}^3/V\mathbb{Z}^3$ ? And how do we exhibit the Dhar bijection between the sandpile states and spanning trees in this case?
I got a list of stable configurations by drawing all marking the root with x and writing down all the possible configurations
1-0 1-1 0-1 1-1
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x-1 x-0 x-1 x-1
My question is how do we match these configuations with the spanning trees above?