A 5% random perturbation of Z_p (30 000 realizations, at the example of Z_p from Boldyrev 2002, p = 1 to p = 5) causes smearing of co-dimensions C derived from Eq. (20) (left panel). A double peak distribution results if C is computed from Eq. (21) (middle panel). The analytical value C = 1 is shown as a red line, green lines in the left panel indicate where the 2/3 largest values in the histogram are located. In the right panel, the 2D histogram (log10 contours, spacing 0.5, spanning three orders of magnitude) of co-dimensions from perturbed Z_p (500 000 realizations) is over-plotted on the co-dimension C = Δ/(1 − β) (color coded) in the Δ-β plane (Eq. (21)). Two sharp peaks can be distinguished, at Δ ≈ 0.4 and β ≈ 0.1 (C ≈ 0.44), as well as at Δ ≈ 1.0 and β ≈ 0.5 (C ≈ 1.9). The red dot is the analytical value by Boldyrev (2002), Δ = 2/3, β = 1/3, C = 1.