Lévy Flights

Power-law jump lengths create "clustering with long jumps" — compare to Gaussian walks

Lévy: step ~ |r|^{-(1+α)}
α=2: Gaussian (CLT)
α<2: heavy-tailed, infinite variance
α<1: infinite mean jump length

P(|X|>x) ~ x^{-α}
MSD ~ t^{2/α} (superdiffusion)
Lévy flights are random walks where step lengths follow a power-law distribution P(ℓ) ~ ℓ^{-(1+α)}. For α<2, variance is infinite; for α<1, even the mean step length is infinite. The characteristic pattern: long periods of local exploration punctuated by rare giant leaps. Unlike Gaussian walks (CLT applies), Lévy walks are superdiffusive: ⟨r²⟩ ~ t^{2/α} grows faster than linear. Found in nature: foraging patterns of albatrosses, sharks, and even human travel. The Lévy stable distributions are the basin of attraction for sums of heavy-tailed random variables — the generalized CLT.