Schematic diagram of the Euclidean volume scaling of the diffusive avalanche boundaries, visualized as circles or spheres in the three Euclidean space dimensions S = 1,2,3. The Euclidean length scale x of subcubes decreases by a factor 2 in each step (x_i = 2 ^− i,i = 0,1,2), while the number of subcubes increases by n_i = (2ⁱ)^S, defining a probability of for each avalanche size with size x_i.