Heavy-tailed step lengths break normal diffusion scaling
2D walker trajectories
MSD(t) vs t — normal vs anomalous
α = ? | Expected MSD ~ t? | Regime: ?
A Lévy walk draws step lengths from a heavy-tailed distribution P(ℓ) ~ ℓ^{−α−1} for large ℓ. Unlike Lévy flights (instantaneous jumps), in a Lévy walk the particle moves continuously at constant speed — step duration equals step length.
The mean squared displacement scales as ⟨r²⟩ ~ t^ν:
• α > 2: normal diffusion ν = 1 (finite variance)
• 1 < α < 2: superdiffusion ν = 4−α > 1 (rare long steps dominate)
• α < 1: ballistic-like ν = 2 (very heavy tail)
Lévy walks appear in animal foraging (albatross, sharks), human mobility, light in random media, and anomalous transport in plasma. The Lévy foraging hypothesis (Viswanathan 1999) proposes that α ≈ 2 is optimal for random search when prey is sparse.
Generation: for power-law P(ℓ) ~ ℓ^{−α−1}, use inverse transform: ℓ = ℓ_min · U^{−1/α} where U ~ Uniform(0,1). The walker moves in a random direction for duration ℓ, then picks a new direction.