EXPLORABLES

This explorable illustrates the properties of a class of random walks known as Lévy flights.
To get the most out of this explorable, you may want to check out the explorable **Albert & Carl Friedrich** on ordinary random walks and diffusion first.

Lévy flights, unlike ordinary random walks, are superdiffusive and scale-free, their paths have a self-similar structure. They play a role in numerous systems in physics but also in ecology and other fields.

* Press play* and keep on reading.

## This is how it works

The two-dimensional Lévy flights here all start at the origin and perform a sequence of random steps. At every iteration $n=1,2,3,…$ their position $\mathbf{R}_{n}=(X_n,Y_n)$ is updated by a displacement $\mathbf{r}_n=(x_n,y_n)$ so that

$$
\mathbf{R}*{n+1}=\mathbf{R}*{n}+\mathbf{r}_n.
$$

At every step a random direction is chosen (uniformly) and a step-length $r$ is drawn from a probability density that follows an inverse power-law

$$ p(r)\propto\frac{1}{r^{1+\mu}} $$

for $r>r_0$ ($r_0$ is some minimal step length). So short jumps are more frequent than long ones. The exponent $0<\mu$ is the Lévy exponent and a defining feature of the geometry of the trajectories. * You can pick different values* for $\mu$ in the control panel. You can also

*to be generated during a run (A run automatically stops after 1000 steps).*

**choose the number of walks**As the walks proceed, the * path*, the

*and the*

**visited locations***are displayed for each walk.*

**current position***to display or hide locations, current position, and paths.*

**You can use the toggles**## Observe this

When you choose a Lévy flight, say for $\mu=1$, you will see that a walker remains in some location for some time making many short jumps. But every now and then a long jump occurs, taking the walker to a distance place. As we zoom out, longer and longer jumps occur, turning the path of the walker into a self-similar structure. * When you hide the path and just look at the visited locations*, this feature is best visible. For smaller exponents the

*“dust”*of visited locations becomes more and more dilute. It turns out that the visited locations form a random fractal of dimension $\mu$ if $\mu<2$.

## What’s up with the central limit theorem?

You may rightfully ask why Lévy flights appear to have such different geometry than paths of ordinary random walks / diffusion processes? After all, does not the central limit theorem dictate that asymptotically these walks should resemble trajectories of ordinary diffusion as illustrated in the explorable **Albert & Carl Friedrich** (check it out, if you haven’t done so already)? Well, for Lévy flights, one of the conditions required for the central limit theorem is violated for $\mu\leq 2$, namely that the variance of the step length $r$ is finite. In fact, for the power-law $p(r)$ above the variance is infinite when $\mu\leq 2$.

## Superdiffusion

As a consequence of this, the position after $n$ steps no longer exhibits the typical scaling $R_n\sim \sqrt{n}$. Instead, Lévy flights are superdiffusive and their typical distance from the origin scales as

$$ R_n \sim n^{\frac{1}{\mu}} $$

Lévy flights, just like ordinary random walks have a limiting distribution, only that it is no longer a Gaussian function. One can show that for $\mu < 2 $ the scaled variable $\mathbf{z}_{n} = \mathbf{R}*n /n^{\frac{1}{\mu}}$ has a probability density that approaches a function $L*\mu(z)$ given by

$$ L_\mu(z)=\int d\mathbf{k}e^{-|\mathbf{k}|^\mu-i\mathbf{k}\mathbf{x}}. $$

When $\mu>2$ the variance of the single steps is finite and everything is back to normal.