Continuous Abelian Sandpile Model in Two Dimensional Lattice

TL;DR

This paper introduces a continuous version of the Abelian Sandpile Model, maintaining key properties while enabling exploration of effects like small mass influence and boundary conditions.

Contribution

It presents a continuous-height sandpile model that is mathematically equivalent to ASM but allows investigation of new phenomena.

Findings

Model can be mapped to ASM, sharing its properties

Allows analysis of small mass effects on height probabilities

Enables study of different boundary conditions

Abstract

We define a new version of sandpile model which is very similar to Abelian Sandpile Model (ASM), but the height variables are continuous ones. With the toppling rule we define in our model, we show that the model can be mapped to ASM, so the general properties of the two models are identical. Yet the new model allows us to investigate some problems such as effect of very small mass on the height probabilities, different boundary conditions, etc.