Random walk

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Example of eight random walks in one dimension starting at 0. The plot shows the current position on the line (vertical axis) versus the time steps (horizontal axis).
Example of eight random walks in one dimension starting at 0. The plot shows the current position on the line (vertical axis) versus the time steps (horizontal axis).

A random walk, sometimes called a "drunkard's walk," is a formalization in mathematics, computer science, and physics of the intuitive idea of taking successive steps, each in a random direction. For example, the path traced by a molecule as it travels in a liquid or a gas is a random walk.

Notably, the drunkard's random walk from a lamp post returns him to it again inevitably (see two-dimensional random walk below).

Contents

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.

A one-dimensional random walk takes place on a line. So, you start at zero, and at each step move by a fixed amount along one of the two directions from the current point, with the direction being chosen randomly.

The average straight-line distance (average amount moved away from zero) between start and finish points of a one-dimensional random walk of n steps is on the order of \sqrt{n}, or more precisely, its asymptote converges to \sqrt{2 n \over \pi} \approx 0.8 \sqrt{n}. This can be described with an example. Say you have a coin. If it lands on heads, you move one to the right on the number line. If it lands on tails, you move one to the left. So after five flips, you have the possibility of landing on 1, -1, 3, -3, 5, or -5. You can land on 1 by flipping three heads and two tails in any order. There are 10 possible ways of landing on 1. Similarly, there are 10 ways of landing on -1 (by flipping three tails and two heads), 5 ways of landing on 3 (by flipping four heads and one tail), 5 ways of landing on -3 (by flipping four tails and one head), 1 way of landing on 5 (by flipping five heads), and 1 way of landing on -5 (by flipping five tails). See the figure below for an illustration of this example.

Five flips of a fair coin
Five flips of a fair coin

As expected of a random walk with equally probable outcomes, the expected value will come out to zero. (The average distance moved.) This can be expressed using the example above: 1×(10/32) + -1×(10/32) + 3×(5/32) + -3×(5/32) + 5×(1/32) + -5×(1/32) = 0.

So if we want to know the average distance moved away from zero in either direction, we can use the square root of the squares. When we square the expected values, they will all turn out positive so that they cannot cancel each other out. This is called the root mean square. Using the example above, we have the squared expression: (1)²×(10/32) + (-1)²×(10/32) + (3)²×(5/32) + (-3)²×(5/32) + (5)²×(1/32) + (-5)²×(1/32) = 5. Taking the square root of the answer, we find that the average distance moved away from zero after 5 flips is the square root of 5. This gives us the generalization that the average distance after n steps is \sqrt{n} times the step length exactly.


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 in one dimension, every point in the domain will almost surely be crossed an infinite number of times. [In two dimensions, this is equivalent to the statement that any line will be crossed an infinite number of times.] 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, and it will almost surely, at some time, reach 0 and the game will be over.

The expected number of steps until a one dimensional random walk goes up to b or down to -a is ab. The probability that the random walk will go up to b steps before going down a steps is a \over a+b

Pascal's triangle also shows up in the analysis of the probabilities of one-dimensional random walks. Examine the probabilities. (If we factor out the 1/2N, there is a pattern in these probabilities.) At zero turns, the only possibility will be to remain at zero. However, at one turn, you can move either to the left or the right of zero, meaning there is one chance of landing on -1 or one chance of landing on 1. At two turns, you examine the turns from before. If you had been at 1, you could move to 2 or back to zero. If you had been at -1, you could move to -2 or back to zero. So there is one chance of landing on -2, two chances of landing on zero, and one chance of landing on 2. If you continue the analysis of probabilities, you can see Pascal's triangle.

n -5 -4 -3 -2 -1 0 1 2 3 4 5
f0(n) 1
2f1(n) 1 1
22f2(n) 1 2 1
23f3(n) 1 3 3 1
24f4(n) 1 4 6 4 1
25f5(n) 1 5 10 10 5 1

Random walk in two dimensions.
Random walk in two dimensions.
Random walk in two dimensions with more, and smaller, steps. In the limit, for very small steps, one obtains the Brownian motion.
Random walk in two dimensions with more, and smaller, steps. In the limit, for very small steps, one obtains the Brownian motion.

Imagine now a drunkard walking randomly in a city. The city is realistically infinite and arranged in a square grid, and at every intersection, the drunkard 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 (almost surely). This is the high dimensional equivalent of the level crossing problem discussed above. However, in dimensions 3 and above, this no longer holds. In other words, a drunk bird might forever wander the sky, never finding its nest. The formal terms to describe this phenomenon is that a random walk in dimensions 1 and 2 is recurrent, while in dimension 3 and above it is transient. This was proven by Pólya in 1921, and is discussed in a section of Markov Chains available online.

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 one 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.

Three random walks in three dimensions.
Three random walks in three dimensions.

Assume now that our city is no longer a perfect square grid. 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.

This characterization of recurrence and transience is very useful, and specifically it allows us 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). This property has important consequences.

Starting in the 1980s, 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.

Simulated steps approximating Brownian motion in two dimensions.
Simulated steps approximating 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 L/ε² 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 [1]. In two dimensions, the average number of points the same random walk has on the boundary of its trajectory is r4 / 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 2000 (Science, 2000).

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, for example, 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:

\sigma^2 = \frac{t}{\delta t}\,\epsilon^2, where t is the time elapsed since the start of the random walk, ε is the size of a step of the random walk, and δt 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 Green's function of the diffusion equation is:

\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:

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 \vec R that links the two ends of the random walk, in 3D. The variance associated to each component Rx, Ry or Rz is only one third of this value (still in 3D).

There are 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

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.
  • In vision science, fixational eye movements are well described by a random walk.
  • In psychology, random walks explain accurately the relation between the time needed to make a decision and the probability that a certain decision will be made. (Nosofsky, 1997)
  • 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.
  • In wireless networking, random walk is used to model node movement.
  • Bacteria engage in a biased random walk.
  • Random walk is used to model gambling.
  • During World War II a random walk was used to model the distance that an escaped prisoner of war would travel in a given time.

A one-dimensional random walk can also be looked at as a Markov chain whose state space is given by the integers i=0,\pm 1,\pm 2,..., for some number \,0 < p < 1, \,P_{i,i+1}=p=1-P_{i,i-1}. We can call it a random walk because we may think of it as being a model for an individual walking on a straight line who at each point of time either takes one step to the right with probability p or one step to the left with probability 1 − p.

A random walk is a simple stochastic process.

  1. ^ Hence the drunkard's random walk would eventually cover all of the city streets (2 Euclidean dimensions) and he will eventually return home, whereas the bird taking a 'random walk' flight through the air (3 Euclidean dimensions) will not cover all space, and will not return to their starting point.

Chapter 3 of this book contains a thorough discussion of random walks, including advanced results, using only elementary tools.
  • Barry D. Hughes (1996), Random walks and random environments, Oxford University Press. ISBN 0-19-853789-1
  • Gregory Lawler (1996), Intersection of random walks, Birkhäuser Boston. ISBN 0-8176-3892-X
  • Gregory Lawler, Conformally Invariant Processes in the Plane, http://www.math.cornell.edu/~lawler/book.ps
  • Neal Madras and Gordon Slade (1996), The Self-Avoiding Walk, Birkhäuser Boston. ISBN 0-8176-3891-1
  • James Norris (1998), Markov Chains, Cambridge University Press. ISBN 0-5216-3396-6
  • Springer Pólya (1921), "Über eine Aufgabe der Wahrscheinlichkeitstheorie betreffend die Irrfahrt im Strassennetz", Mathematische Annalen, 84(1-2):149–160, March 1921.
  • Pal Révész (1990), Random walk in random and non-random environments, World Scientific Pub Co. ISBN 981-02-0237-7
  • Wolfgang Woess (2000), Random walks on infinite graphs and groups, Cambridge tracts in mathematics 138, Cambridge University Press. ISBN 0-521-55292-3

Retrieved from "http://en.wikipedia.org/wiki/Random_walk"
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