Lévy flight

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.

Figure 1. An example of 1000 steps of a Lévy flight in two dimensions. The origin of the motion is at [0,0] and the x and y components of each step are independent and distributed according to a symmetric, centered Lévy distribution with c = 1 and α = 1.2. 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 in two dimensions. The origin of the motion is at [0, 0] and the x and y components of each step are independent and are distributed according to a symmetric, centered Lévy distribution with c = 1 and α = 2 which is equivalent to a normal distribution with a variance of 2 and a mean of zero.

See also

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

External links

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