Left: Excursion set perimeter length, W₁. Right: Euler characteristic – measured (red points and error bars), mean-field expectation value (dashed black lines) and BMF with α = 1.5 (dashed blue lines) and α = 1.43 (solid blue lines). The error bars represent the standard deviation, and the points are the sample means. The top to bottom panels are the cases ξ₀ = 0.08, 0.88, 2.47, respectively, with the parameters Δ = 0.25, R = 1, L = 4000 × Δ, α = 3/2, and N_pts = 40 × 2000². Once again the agreement is excellent, especially at the high-density end. Note that the BMF prediction for χ (right panels) is beyond the scope of this paper.