Scaling of waves in the Bak-Tang-Wiesenfeld sandpile model

TL;DR

This paper analyzes wave distributions in the Bak-Tang-Wiesenfeld sandpile model, revealing power law behaviors, critical exponents, and confirming the upper critical dimension as 4 with logarithmic corrections.

Contribution

It provides the first comprehensive numerical and analytical study of wave distributions, establishing their scaling laws and critical exponents across different dimensions.

Findings

Wave distributions follow power law asymptotics.

Critical exponents are determined for various wave properties.

Upper critical dimension is confirmed as 4 with logarithmic corrections.

Abstract

We study probability distributions of waves of topplings in the Bak-Tang-Wiesenfeld model on hypercubic lattices for dimensions D>=2. Waves represent relaxation processes which do not contain multiple toppling events. We investigate bulk and boundary waves by means of their correspondence to spanning trees, and by extensive numerical simulations. While the scaling behavior of avalanches is complex and usually not governed by simple scaling laws, we show that the probability distributions for waves display clear power law asymptotic behavior in perfect agreement with the analytical predictions. Critical exponents are obtained for the distributions of radius, area, and duration, of bulk and boundary waves. Relations between them and fractal dimensions of waves are derived. We confirm that the upper critical dimension D_u of the model is 4, and calculate logarithmic corrections to the scaling behavior of waves in D=4. In addition we present analytical estimates for bulk avalanches in dimensions D>=4 and simulation data for avalanches in D<=3. For D=2 they seem not easy to interpret.