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

We study probability distributions of waves of topplings in the Bak-Tang-Wiesenfeld model on hypercubic lattices for dimensions D greater than or equal to 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 greater than or equal to 4 and simulation data for avalanches in D greater than or equal to 3. For D = 2 they seem not easy to interpret.