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" distribution.

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.

A heavy-tailed distribution is a probability distribution which falls to zero as 1/|x|α+1 where 0 < α < 2 and therefore has an infinite variance. According to the central limit theorem, if the distribution were to have a finite variance, then after a large number of steps, the distance from the origin of the random walk would tend to a normal distribution. In contrast, if the distribution is heavy-tailed, then after a large number of steps, the distance from the origin of the random walk will tend to a Lévy distribution. Lévy flight is part of a class of Markov processes.

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.

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 and biology.