The divisible sandpile with heavy-tailed variables

TL;DR

This paper investigates the divisible sandpile model with heavy-tailed initial configurations, establishing conditions for stabilization and showing the odometer's scaling limit is an alpha-stable distribution.

Contribution

It extends previous work by identifying stabilization conditions and characterizing the odometer's limit as an alpha-stable distribution for heavy-tailed inputs.

Findings

Conditions for stabilization and non-stabilization on infinite graphs.

The odometer's scaling limit is an alpha-stable random distribution.

Extension of prior results to heavy-tailed initial configurations.

Abstract

This work deals with the divisible sandpile model when an initial configuration sampled from a heavy-tailed distribution. Extending results of Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient conditions for stabilization and non-stabilization on infinite graphs. We determine furthermore that the scaling limit of the odometer on the torus is an $\alpha$-stable random distribution.