Left: Two-dimensional Lévy flight probability distribution function. Right: difference with respect to the MF with parameters α = 3/2, ξ₀ = 0.08, 0.88, 2.47 (top-bottom panels), L = 4000 × Δ (box width), N_pts = 40 × 2000² (number of steps), Δ = 0.25 (resolution), and R = 1 (smoothing scale). The error bars show the standard deviation in the left panels and the error on the mean for 100 realisations in the right panels. The points are the sample means. The corresponding standard deviations of the fields are given by ${\sigma }_0=\sqrt{{\xi }_0}=0.28,0.94,1.57$ (top-bottom). The BMF prediction agrees well with the simulations even in the non-perturbative, non-Gaussian regime.