Lévy flight

A Lévy flight, named for French mathematician Paul Lévy, is a random walk in which the step-lengths have a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions.

The term "Lévy flight" was coined by Benoît Mandelbrot, who used this for one specific definition of the distribution of step sizes. He used the term Cauchy flight for the case where the distribution of step sizes is a Cauchy distribution, and Rayleigh flight for when the distribution is a normal distribution (which is not an example of a heavy-tailed probability distribution).

Later researchers have extended the use of the term "Lévy flight" to include cases where the random walk takes place on a discrete grid rather than on a continuous space.

A Lévy flight is a random walk in which the steps are defined in terms of the step-lengths, which have a certain probability distribution, with the directions of the steps being isotropic and random.

The particular case for which Mandelbrot used the term "Lévy flight" is defined by the survivor function (commonly known as the survival function) of the distribution of step-sizes, U, being

Here D is a parameter related to the fractal dimension and the distribution is a particular case of the Pareto distribution. Later researchers allow the distribution of step sizes to be any distribution for which the survival function has a power-like tail

for some k satisfying 1 < k < 3. (Here the notation O is the Big O notation.) Such distributions have an infinite variance. Typical examples are the symmetric stable distributions.

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