A Brownian walker takes Gaussian steps — bell curve, no surprises. A Lévy flight takes heavy-tailed steps — most are short, but every so often comes a huge leap. Drop a Brownian walker in a sparse environment: it stays in a small region. Drop a Lévy walker: rare giant jumps let it explore vastly more space. Sharks, T-cells and — more controversially — albatrosses and human hunter-gatherers have all been reported walking this way.
why fat tails beat the bell curve in nature
A random walk takes steps from some distribution. For Gaussian steps, the Central Limit Theorem says total displacement is also Gaussian — scales as √N (Brownian motion).
Lévy distributions are heavy-tailed: P(|step| > x) ∝ x^(-α) for some stability parameter α ∈ (0, 2). The variance is infinite: rare giant steps dominate. The generalized CLT says sums of such steps converge to α-stable distributions, not Gaussian.
Why animals might walk this way: Viswanathan et al. (1999) argued that when targets are sparse and can be revisited, a walk whose step lengths fall off as ℓ⁻² (α = 1 on the slider) is optimal; when targets are depleted on contact, straight-line search does better. This became the Lévy flight foraging hypothesis. It is contested: a 2020 analysis (Levernier et al., PRL) showed the optimum isn't robust in two or more dimensions.
Reported in (with varying support):
• Marine predators — sharks, tuna, turtles: Lévy-like search in sparse prey fields, Brownian in rich ones (Humphries et al. 2010)
• T-cells hunting for infected cells in the brain: a generalized Lévy walk (Harris et al. 2012)
• Wandering albatrosses: the famous 1996 result was overturned in 2007 (Edwards et al., Nature) — many "long flights" were really time spent sitting on the water. Later GPS work found some individual birds do fly Lévy-like paths. A cautionary tale about power-law claims.
• Human mobility: Brockmann's banknote-tracking study (Where's George data) found Lévy-like statistics over kilometer scales; phone data shows more truncated distributions
α controls fatness: α = 2 is the Gaussian limit (no heavy tail). α → 0⁺ is dominated by single huge jumps. α = 1 is the Cauchy distribution. Many reported ecological fits sit near α ≈ 1 — though fitting power laws to movement data is notoriously treacherous. The "search by lottery" strategy can beat systematic combing in sparse environments; whether nature evolved it on purpose is still argued.