Lévy Flights vs Gaussian Random Walks

Lévy Flights (Lévy 1937): When step sizes are drawn from a power-law distribution P(l)∝l^{−(α+1)} with α<2, the central limit theorem fails — the variance diverges. The resulting random walk is a Lévy flight with self-similar, fractal trajectories and occasional enormous jumps. Mean squared displacement ⟨r²⟩∝t^{2/α} diverges for α<2 (superdiffusion). At α=2, Lévy flights approach Brownian motion. The displacement distribution is a Lévy stable distribution (heavy-tailed, fat) vs. Gaussian for normal walks. Lévy flights appear in animal foraging, financial returns, and anomalous diffusion in disordered media.