Scaling limit of the odometer in divisible sandpiles

TL;DR

This paper proves that the rescaled odometer function in divisible sandpiles converges to the continuum bilaplacian field on the unit torus across all dimensions, establishing a key link between discrete models and continuum fields.

Contribution

It demonstrates the universal convergence of the odometer in divisible sandpiles to the continuum bilaplacian field in any dimension, extending previous conjectures.

Findings

Rescaled odometer converges to the continuum bilaplacian field

Convergence holds in all dimensions

Supports the conjecture relating discrete sandpiles to continuum fields

Abstract

In a recent work Levine et al. (2015) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.