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

We study probability distributions of waves of topplings in the Bak-Tang-Wi esenfeld model on hypercubic lattices for dimensions D greater than or equa l to 2. Waves represent relaxation processes which do not contain multiple toppling events. We investigate bulk and boundary waves by means of their c orrespondence to spanning trees, and by extensive numerical simulations. Wh ile the scaling behavior of avalanches is complex and usually not governed by simple scaling laws, we show that the probability distributions for wave s display clear power-law asymptotic behavior in perfect agreement with the analytical predictions. Critical exponents are obtained for the distributi ons of radius, area, and duration of bulk and boundary waves. Relations bet ween 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 c orrections to the scaling behavior of waves in D = 4. In addition, we prese nt 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 t o 3. For D = 2 they seem not easy to interpret.