A Lévy flight, named after the French mathematician Paul Pierre Lévy, is a type of random walk in which the increments are distributed according to a heavy-tailed probability distribution. Specifically, the distribution used is a power law of the form y = x ^-α where 1 < α < 3 and therefore has an infinite variance.

Lévy flights are Markov processes. After a large number of steps, the distance from the origin of the random walk tends to a stable distribution.

Two-dimensional Lévy flights were described by Benoît Mandelbrot in The Fractal Geometry of Nature. The exponential scaling of the step lengths gives Lévy flights a scale invariant property, and they are used to model data that exhibits clustering.

This method of simulation stems heavily from the mathematics related to chaos theory and is useful in stochastic measurement and simulations for random or pseudo-random natural phenomena. Examples include earthquake data analysis, financial mathematics, cryptography, signals analysis as well as many applications in astronomy, biology, and physics.

When sharks and other ocean predators can’t find food, they abandon Brownian motion, the random motion seen in swirling gas molecules, for Lévy flight — a mix of long trajectories and short, random movements found in turbulent fluids. Researchers analyzed over 12 million movements recorded over 5,700 days in 55 radio-tagged animals from 14 ocean predator species in the Atlantic and Pacific Oceans, including silky sharks, yellowfin tuna, blue marlin and swordfish. The data showed that Lévy flights interspersed with Brownian motion can describe the animals' hunting patterns.^[1][2]

Figure 1. An example of 1000 steps of a Lévy flight in two dimensions. The origin of the motion is at [0,0], the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e. stable) distribution with α=1 and β=0 (i.e. a Cauchy distribution). Note the presence of large jumps in location compared to the Brownian motion illustrated in Figure 2.

Figure 2. An example of 1000 steps of an approximation to a Brownian motion type of Lévy flight in two dimensions. The origin of the motion is at [0, 0], the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e. stable) distribution with α=2 and β=0 (i.e. a Normal distribution

See also

• Monte Carlo method
• Pseudo-random number
• Chaos theory
• Random walk
• Fourier transform
• Crystallography
• Geology
• Astronomy
• Fat tail
• Lévy process
• Fractional quantum mechanics

References

External links

• A comparison of the paintings of Jackson Pollock to a Lévy flight model