# Random walk

A random walk is a trajectory that is created from successive random steps. Depending on the dimension of the space the walk is performed in and the definition of randomness used for each step, different kinds of walks can be formulated.

Different spaces for a random walk include the one-dimensional space of integers (e.g. successively flipping a coin with values of $\pm 1$ and adding these up or the Fermi estimation), the plane with real coordinates (e.g. the drunkard's walk), or 3D euclidean space (e.g. Levy Flight or the Wiener process). More uncommon spaces include graphs (see the overview here) or groups in the mathematical sense (a short introduction can be found here).

Steps can be performed at defined time intervals or at random times, with a defined or random step length and each random component can be modified based on previous steps.

In the context of soft matter, the two most important applications of the concept of random walks are diffusion limited aggregation and the modeling of polymers as freely-jointed chains.

For a polymer chain undergoing a completely ramdom walk the end-to-end length $R$ scales with the number of segments $N$ like

$R \propto \sqrt{N}$