Random walk
A random walk, sometimes denoted RW, is a mathematical formalization of a trajectory that consists of taking successive random steps.  The results of random walk analysis have been applied to computer science, physics, ecology, economics and a number of other fields as a fundamental model for random processes in time.  For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler can all be modeled as random walks.
Specific cases or limits of random walks include the drunkard's walk and Levy flight. Random walks are related to the diffusion models and are a fundamental topic in discussions of Markov processes.  Several properties of random walks, including dispersal distributions, first-passage times and encounter rates, have been extensively studied.
Various different types of random walks are of interest. Often, random walks are assumed to be Markov, but other, more complicated walks are also of interest. Some random walks are on graphs, others on the line, in the plane, or in higher dimensions, while some random walks are on groups. Random walks also vary with regards to the time parameter. Often, the walk is indexed by the natural numbers, as in X_0,X_1,X_2,\dots. However, some walks take their steps at random times, and in that case the position X_t is defined for t\ge 0.

# One-dimensional random walk
A particularly elementary and concrete random walk is the random walk on the integers \mathbb Z, which starts at S_0=0 and at each step moves by \pm1 with equal probability. To define this walk formally, take independent random variables Z_1,Z_2,\dots, each of which is 1 with probability 1/2 and -1 with probability 1/2, and set S_n:=\sum_{j=1}^nZ_j. This sequence \{S_n\} is called the simple random walk on \mathbb Z.
This walk can be illustrated as follows. Say you flip a fair 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.
What can we say about the position S_n of the walk after n steps? Of course, it is random, so we cannot calculate it. But we may say quite a bit about its distribution. It is not hard to see that the expectation E(S_n) of S_n is zero. For example, this follows by the additivity property of expectation: E(S_n)=\sum_{j=1}^n E(Z_n)=0.
A similar calculation, using the independence of the random variables Z_n, shows that E(S_n^2)=n. This hints that E|S_n|, the expected translation distance after n steps, should be of the order of \sqrt{n}. In fact,
Suppose we draw a line some distance from the origin of the walk. How many times will the random walk cross the line if permitted to continue walking forever? The following, perhaps surprising theorem is the answer: simple random walk on \mathbb Z will almost surely cross every point an infinite number of times. This result has many names: the level-crossing phenomenon, recurrence or the gambler's ruin. The reason for 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.
If a and b are positive integers, then the expected number of steps until a one dimensional simple random walk starting at 0 first hits b or -a is a\,b.  The probability that this walk will hit b before -a steps is a \over a+b.
Some of the results mentioned above can be derived from properties of Pascal's triangle. The number of different walks of n steps where each step is +1 or -1 is clearly 2^n. For the simple random walk, each of these walks are equally likely. In order for S_n to be equal to a number k it is necessary and sufficient that the number of +1 in the walk exceeds those of -1 by k. Thus, the number of walks which satisfy S_n=k is precisely the number of ways of choosing (n+k)/2 elements from an n element set (for this to be non-zero, it is necessary that n+k be an even number), which is an entry in Pascal's triangle denoted by n \choose (n+k)/2. Therefore, the probability that S_n=k is equal to 2^{-n}{n\choose (n+k)/2}. By representing entries of Pascal's triangle in terms of factorials and using Stirling's formula, one can obtain good estimates for these probabilities for large values of n.
This relation with Pascal's triangle is easily demonstrated for small values of n. 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.
The central limit theorem and the law of the iterated logarithm describe important aspects of the behavior of simple random walk on \mathbb Z.

# Higher dimensions
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. 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 Polya 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.

# Random walk on graphs
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 (an elementary proof can be found in the book by Doyle and Snell):
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 if the graph is connected.
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 Poincare 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 Dave Bayer and 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
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 , one needs to take a walk of length L/2 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 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 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:
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:
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:
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 R_x, R_y or R_z is only one third of this value (still in 3D).

# Self-interacting random walks
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
- The self-avoiding walk. See the Madras and Slade book.
- The loop-erased random walk. See the two books of Lawler.
- The reinforced random walk. See the review by Robin Pemantle.
- The exploration process.

# Applications
The following are the applications of random walk:
- In economics, the "random walk hypothesis" is used to model shares prices and other factors. Empirical studies found some deviations from this theoretical model, especially in short term and long term correlations. See share prices.
- In population genetics, random walk describes the statistical properties of genetic drift
- In physics, random walks are used as simplified models of physical Brownian motion and the random movement of molecules in liquids and gases. See for example diffusion-limited aggregation.
- In mathematical ecology, random walks are used to describe individual animal movements, to empirically support processes of biodiffusion, and occasionally to model population dynamics.
- Also in physics, random walks and some of the self interacting walks play a role in quantum field theory.
- In polymer physics, random walk describes an ideal chain. It is the simplest model to study polymers.
- In other fields of mathematics, random walk is used to calculate solutions to Laplace's equation, to estimate the harmonic measure, and for various constructions in analysis and combinatorics.
- In computer science, random walks are used to estimate the size of the Web. In the World Wide Web conference-2006, bar-yossef et.al. published their findings and algorithms for the same. (This was awarded the best paper for the year 2006).
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.

# Probabilistic interpretation
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_{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.