2D Trajectory
Position Distribution (x-axis projection)
MSD vs Time (log-log)
Heavy-tailed random walks, α-stable distributions, and super-diffusive scaling — explore non-Gaussian random processes that govern animal foraging, financial returns, and turbulent transport
An α-stable distribution generalizes the normal distribution. Its characteristic function is φ(t)=exp(−|σt|^α). Heavy tails follow the power law P(|l|>x) ~ x^(−α).
Step length follows P(l) ∝ l^(−1−α). When α<2 the variance is infinite. Rare but enormous jumps dominate the trajectory and break the classical central limit theorem.
Chambers-Mallows-Stuck (1976) generates symmetric α-stable samples from a uniform U∈(−π/2,π/2) and an exponential W∈Exp(1): X = sin(αU)/cos(U)^(1/α) · [cos((1−α)U)/W]^((1−α)/α).
The mean squared displacement grows as ⟨x²⟩ ~ t^(2/α). Brownian (α=2) gives linear t; Cauchy (α=1) gives quadratic t². For α<2 super-diffusion dominates — the walker explores space far faster than ordinary diffusion theory predicts.
The classical CLT requires finite variance. With heavy tails (α<2) the rescaled sum n^(−1/α)Σξ_i still converges — but to an α-stable distribution, not Gaussian. α-stable laws are the only attractors for power-law summands.
Lévy flight trajectories are self-similar: zooming in by factor b on time and b^(1/α) on space yields a statistically identical picture. Their fractal dimension equals α.
Albatrosses, sharks, marine predators and even T-cells move in Lévy patterns when food is sparse: short local searches punctuated by long relocation jumps. α≈2 is optimal for sparse-resource search (Viswanathan, 1996).
Asset returns exhibit fat tails with α∈(1.5,1.8), explaining 'black swan' events that Gaussian Black-Scholes models underestimate. Mandelbrot (1963) proposed Lévy-stable laws for cotton prices.
Superdiffusion appears in turbulent flows, plasma transport, photon transport in cold atoms, and molecular motion in disordered media. Continuous-Time Random Walks (CTRW) connect Lévy flights to fractional Fokker–Planck equations.