Effect of Noise on Patterns Formed by Growing Sandpiles

TL;DR

This paper investigates how different types of noise affect the formation and robustness of patterns in growing sandpiles, revealing that some noise types preserve patterns while others disrupt them entirely.

Contribution

It provides a comprehensive analysis of noise effects on sandpile patterns and introduces a variational approach to pattern selection in these models.

Findings

Patterns are robust to small randomness in addition points and initial configurations.

Disorder in lattice connectivity can partially disrupt pattern formation.

Stochastic toppling rules destroy asymptotic patterns even with weak noise.

Abstract

We consider patterns generated by adding large number of sand grains at a single site in an abelian sandpile model with a periodic initial configuration, and relaxing. The patterns show proportionate growth. We study the robustness of these patterns against different types of noise, \textit{viz.}, randomness in the point of addition, disorder in the initial periodic configuration, and disorder in the connectivity of the underlying lattice. We find that the patterns show a varying degree of robustness to addition of a small amount of noise in each case. However, introducing stochasticity in the toppling rules seems to destroy the asymptotic patterns completely, even for a weak noise. We also discuss a variational formulation of the pattern selection problem in growing abelian sandpiles.