# A stochastic variant of the abelian sandpile model

## TL;DR

This paper introduces a stochastic variant of the abelian sandpile model (SSP), exploring its mathematical properties, physical behavior, and potential as a simplified model for understanding the LLL algorithm in lattice cryptography.

## Contribution

The paper develops a basic theoretical framework for SSP, a stochastic extension of ASM, and investigates its properties and potential applications in cryptography.

## Key findings

- SSP shares mathematical properties with ASM
- SSP exhibits different physical behavior and steady state shape
- Numerical study of SSP's behavior supports its theoretical analysis

## Abstract

We introduce a natural stochastic extension, called SSP, of the abelian sandpile model(ASM), which shares many mathematical properties with ASM, yet radically differs in its physical behavior, for example in terms of the shape of the steady state and of the avalanche size distribution. We establish a basic theory of SSP analogous to that of ASM, and present a brief numerical study of its behavior. Our original motivation for studying SSP stems from its connection to the LLL algorithm established in another work by the authors [5]. The importance of understanding how LLL works cannot be stressed more, especially from the point of view of lattice-based cryptography. We believe SSP serves as a tractable toy model of LLL that would help further our understanding of it.