Scaling limit of the sandpile identity element on the Sierpinski gasket

TL;DR

This paper studies the behavior of the sandpile group's identity element on approximations of the Sierpinski gasket, showing it converges to a constant function in the limit, and extends the results to various configurations.

Contribution

It provides the first analysis of the scaling limit of the sandpile identity element on fractal structures like the Sierpinski gasket, including generalizations for different sink vertices.

Findings

The identity elements converge to a constant function with value 4.

The convergence occurs in the weak* sense.

Results are extended to various sink configurations.

Abstract

We investigate the identity element of the sandpile group on finite approximations of the Sierpinski gasket with normal boundary conditions and show that the sequence of piecewise constant continuations of the identity elements on SG_n converges in the weak* sense to the constant function with value 4 on the Sierpinski gasket SG. We then generalize the proof to a wider range of functions and obtain the scaling limit for the identity elements with different choices of sink vertices.