Direct Evidence for Conformal Invariance of Avalanche Frontier in Sandpile Models

TL;DR

This paper provides numerical evidence that avalanche frontiers in Abelian and Zhang sandpile models are conformally invariant and can be described by SLE with diffusivity 2, linking them to loop erased random walks.

Contribution

The study demonstrates conformal invariance of avalanche frontiers in sandpile models using SLE, a novel application in this context.

Findings

Avalanche frontiers in ASM are conformally invariant and described by SLE with κ=2.

Fractal dimension and Schramm's formula support the SLE description.

Similar properties are observed in Zhang's sandpile model.

Abstract

Appreciation of Stochastic Loewner evolution (SLE$_\kappa$), as a powerful tool to check for conformal invariant properties of geometrical features of critical systems has been rising. In this paper we use this method to check conformal invariance in sandpile models. Avalanche frontiers in Abelian sandpile model (ASM) are numerically shown to be conformally invariant and can be described by SLE with diffusivity $\kappa=2$. This value is the same as value obtained for loop erased random walks (LERW). The fractal dimension and Schramm's formula for left passage probability also suggest the same result. We also check the same properties for Zhang's sandpile model.