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

TL;DR

This paper investigates the structure of the sandpile identity element on Sierpinski gasket graphs, revealing its scaling limit behavior and decomposition related to graph distance, contributing to understanding complex fractal network dynamics.

Contribution

It introduces a novel decomposition of the sandpile identity element on Sierpinski graphs, linking it to graph distance and its scaling limit behavior.

Findings

Second-order term converges to the path distance to the nearest corner

Decomposition of the identity into constant and Laplacian of graph distance

Scaling limit behavior characterized for Sierpinski gasket graphs

Abstract

We consider the identity of the abelian sandpile group of finite approximation graphs of the Sierpinski gasket, and we show that the second-order term in the scaling limit converges to the path distance to the nearest corner on the Sierpinski 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.