On the structure of the sandpile identity element on Sierpiński gasket graphs

We consider the identity of the abelian sandpile group of finite approximation graphs of the Sierpiński gasket, and we show that the second-order term in the scaling limit converges to the path distance to the nearest corner on the Sierpiński gasket. The proof relies on a decomposition of the identity of the sandpile group into the sum of a constant function and the Laplacian of the graph distance on the approximating graphs.