# Random walk

In mathematics and physics, a random walk is a formalization of the intuitive idea of taking successive steps, each in a random direction. A random walk is a simple stochastic process.

A random walk is sometimes called a "drunkard's walk". Drunkard's Walk is also the name of a 1960 science fiction novel by Frederik Pohl.

## Properties

The simplest random walk is a path constructed according to the following rules:

• There is a starting point.
• The distance from one point in the path to the next is a constant.
• The direction from one point in the path to the next is chosen at random, and no direction is more probable than another.

The average straight-line distance between start and finish points of a random walk of n steps is on the order of $\displaystyle \sqrt{n}$ , or more precisely, it asymptotes to $\displaystyle \sqrt{2 n \over \pi} \approx 0.8 \sqrt{n}$ . If "average" is understood in the sense of root-mean-square, then the average distance after n steps is higher, but still less than $\displaystyle \sqrt{n}$ times the step length.

Suppose we draw a line some distance from the origin of the walk. How many times will the random walk cross the line? The following, perhaps surprising, theorem is the answer: for any random walk, every point in the domain will be crossed an infinite number of times almost surely. This problem has many names: the level-crossing problem, the recurrence problem or the gambler's ruin problem. The source of the last name is as follows: if you are a gambler with a finite amount of money playing a fair game against a bank with an infinite amount of money, you will surely lose. The amount of money you have will perform a random walk but it will, almost surely, reach at some time 0, and the game would be over.

## Example

Example random walks
Graphs (n,R(n)) of eight random walks starting at 0.

The graph of eight random walks, each starting at zero, are shown here for 100 timesteps. At each time step, they go either one step up or down. As one can see, while they remain clustered around their common origin (the horizontal axis), their average distance to the origin does indeed increase, but more slowly than linearly.

## Higher dimensions

Imagine now a drunkard walking around in the city. The city is infinite and completely ordered, and at every corner he chooses one of the four possible routes (including the one he came from) with equal probability. Formally, this is a random walk on the set of all points in the plane with integer coordinates. Will the drunkard ever get back to his home from the bar? It turns out that he will. This is the high dimensional equivalent of the level crossing problem discussed above. However, the similarity stops here. In dimensions 3 and above, this no longer holds. In other words, a drunk bird might forever wander around, never finding its nest. The formal terms to describe this phenomenon is that random walk in dimensions 1 and 2 is recurrent while in dimension 3 and above it is transient. This was proved by Pólya in 1921.

The trajectory of a random walk is the collection of sites it visited, considered as a set with disregard to when the walk arrived at the point. In 1 dimension, the trajectory is simply all points between the minimum height the walk achieved and the maximum (both are, on average, on the order of √n). In higher dimensions the set has interesting geometric properties. In fact, one gets a discrete fractal, that is a set which exhibits stochastic self-similarity on large scales, but on small scales one can observe "jugginess" resulting from the grid on which the walk is performed. The two books of Lawler referenced below are a good source on this topic.

## Random walk on graphs

Assume now that our city is no longer orderly. When our drunkard reaches a certain junction he picks between the various available roads with equal probability. Thus, if the junction has seven exits the drunkard will go to each one with probability one seventh. This is a random walk on a graph. Will our drunkard reach his home? It turns out that under rather mild conditions, the answer is still yes. For example, if the lengths of all the blocks are between a and b (where a and b are any two finite positive numbers), then the drunkard will, almost surely, reach his home. Notice that we do not assume that the graph is planar, i.e. the city may contain tunnels and bridges. One way to prove this result is using the connection to electrical networks. Take a map of the city and place a one ohm resistor on every block. Now measure the "resistance between a point and infinity". In other words, choose some number R and take all the points in the electrical network with distance bigger than R from our point and wire them together. This is now a finite electrical network and we may measure the resistance from our point to the wired points. Take R to infinity. The limit is called the resistance between a point and infinity. It turns out that the following is true:

Theorem: a graph is transient if and only if the resistance between a point and infinity is finite. It is not important which point is chosen.

In other words, in a transient system, one only needs to overcome a finite resistance to get to infinity from any point. In a recurrent system, the resistance from any point to infinity is infinite.

It also turns out that this characterization of recurrence and transience is very useful, and specifically it allows to analyze the case of a city drawn in the plane with the distances bounded.

A random walk on a graph is a very special case of a Markov chain. Unlike a general Markov chain, random walk on a graph enjoys a property called time symmetry or reversibility. Roughly speaking, this property, also called the principle of detailed balance, means that the probabilities to traverse a given path in one direction or in the other have a very simple connection between them (if the graph is regular, they are just equal). It turns out that this property has important consequences.

Starting from the 80s, much research has gone into connecting properties of the graph to random walks. In addition to the electrical network connection described above, there are important connections to isoperimetric inequalities, see more here, functional inequalities such as Sobolev and Poincaré inequalities and properties of solutions of Laplace's equation. A significant portion of this research was focused on Cayley graphs of finitely generated groups. For example, the proof of Persi Diaconis that 7 riffle shuffles are enough to mix a pack of cards (see more details under shuffle) is in effect a result about random walk on the group Sn, and the proof uses the group structure in an essential way. In many cases these discrete results carry over to, or are derived from Manifolds and Lie groups.

A good reference for random walk on graphs is the online book by Aldous and Fill. For groups see the book of Woess. If the graph itself is random, this topic is called "random walk in random environment" — see the book of Hughes.

## Relation to Brownian motion

File:BrownianMotion.png
An example of 1000 simulated steps of Brownian motion in two dimensions.

Brownian motion is the scaling limit of random walk in dimension 1. This means that if you take a random walk with very small steps you get an approximation to Brownian motion. To be more precise, if the step size is ε, one needs to take a walk of length L2 to approximate a Brownian motion of length L. As the step size tends to 0 (and the number of steps increased comparatively) random walk converges to Brownian motion in an appropriate sense. Formally, if B is the space of all paths of length L with the maximum topology, and if M is the space of measure over B with the norm topology, then the convergence is in the space M. Similarly, Brownian motion in several dimensions is the scaling limit of random walk in the same number of dimensions. Note that Brownian motion in the present article refers to the mathematical definition of the term, rather than the actual physical phenomenon of a minute particle diffusing in a fluid.

A random walk is a discrete fractal, but Brownian motion is a true fractal, and there is a connection between the two. For example, take a random walk until it hits a circle of radius r times the step length. The average number of steps it performs is r2. This fact is the discrete version of the fact that Brownian motion is a fractal of Hausdorff dimension 2. In two dimensions, the average number of points the same random walk has on the boundary of its trajectory is $\displaystyle r^{4/3}$ . This corresponds to the fact that the boundary of the trajectory of Brownian motion is a fractal of dimension 4/3, a fact predicted by Mandelbrot using simulations but proved only in 2001.

Brownian motion enjoys many symmetries random walk does not. For example, Brownian motion is invariant to rotations, but random walk is not, since the underlying grid is not (random walk is invariant to rotations by 90 degrees, but Brownian motion is invariant to rotations by 17 degrees too). This means that in many cases, problems on random walk are easier to solve by translating them to Brownian motion, solving the problem there, and then translating back. On the other hand, some problems are easier to solve with random walks due to its discrete nature.

Random walk and Brownian motion can be coupled, namely manifested on the same probability space in a dependent way that forces them to be quite close. The simplest such coupling is the Skorokhod embedding, but other, more precise couplings exist as well.

The convergence of a random walk toward the brownian motion is controlled by the central limit theorem. For a particle in a known fixed position at t=0, the theorem tells us that after a large number of independent steps in the random walk, the walker's position is distributed according to a normal distribution of total variance:

$\displaystyle \sigma^2 = \frac{t}{\delta t}\,\epsilon^2$ , where t is the time elapsed since the start of the random walk, $\displaystyle \epsilon$ is the size of a step of the random walk, and $\displaystyle \delta t$ is the time elapsed between two successive steps.

This corresponds to the Green function of the diffusion equation that controls the brownian motion, which demonstrates that, after a large number of steps, the random walk converges toward a brownian motion.

In 3D, the variance corresponding to the Greene function of the diffusion equation is:

$\displaystyle \sigma^2 = 6\,D\,t$

By equalizing this quantity with the variance associated to the position of the random walker, one obtains the equivalent diffusion coefficient to be considered for the asymptotic brownian motion toward which the random walk converges after a large number of steps:

$\displaystyle D = \frac{\epsilon^2}{6 \delta t}$ (valid only in 3D)

remark: the two expressions of the variance above correspond to the distribution associated to the vector $\displaystyle \vec R$ that links the two ends of the random walk, in 3D. The variance associated to each compoment $\displaystyle R_x$ , $\displaystyle R_y$ or $\displaystyle R_z$ is only one third of this value (still in 3D).

## Self-interacting random walks

There is a number of interesting models of random paths in which each step depends on the past in a complicated manner. All are more difficult to analyze than the usual random walk — some notoriously so. For example

• The loop-erased random walk. See the two books of Lawler.
• The reinforced random walk.
• The exploration process.

## Applications

In all these cases, random walk is often substituted for Brownian motion.

• In Brain Research, random walks and reinforced random walks are used to model cascades of neuron firing in the brain.
• Random walk can be used to sample from a state space which is unknown or very large, for example to pick a random page off the internet or, for research of working conditions, a random illegal worker in a given country.
• When this last approach is used in computer science it is known as Markov Chain Monte Carlo or MCMC for short. Often, sampling from some complicated state space also allows one to get a probabilistic estimate of the space's size. The estimate of the permanent of a large matrix of zeros and ones was the first major problem tackled using this approach.
• Bacteria engage in a biased random walk.
• And lest we forget, the random walk started its life in modeling of gambling.