RW random walk

A random walk is a mathematical concept that describes a path consisting of a series of random steps taken in different directions. It is often used to model the behavior of systems that exhibit unpredictable movement or fluctuations over time.

A random walk can be thought of as a sequence of steps taken in a particular direction. Each step is determined randomly, and the direction and length of each step are independent of previous steps. The path of a random walk is typically represented in a two-dimensional space, where each step corresponds to a movement in the x or y direction.

There are different types of random walks, and one common type is the one-dimensional random walk (RW). In a one-dimensional random walk, the movement occurs along a single axis, such as a number line. At each step, the walker can move either to the right or to the left, with equal probability.

To illustrate a one-dimensional random walk, let's consider a walker starting at a specific position, denoted by a point on the number line. At each step, the walker randomly chooses to move one unit to the right or one unit to the left. The direction of the step is determined by flipping a fair coin, where heads represent a step to the right, and tails represent a step to the left. This process is repeated for a certain number of steps or until a stopping condition is met.

The walker's position is updated after each step based on the chosen direction. For example, if the walker is initially at position 0 and flips a coin to move to the right, the new position becomes 1. If the next coin flip results in a move to the left, the position would be updated to 0 again. This process continues, and the walker's position evolves according to the random choices made at each step.

One way to analyze the behavior of a random walk is by examining its properties, such as the mean and variance of the walker's position. In a one-dimensional random walk, the mean position remains constant over time, as the probabilities of moving left and right are equal. However, the variance of the position increases with time. This means that the walker becomes increasingly spread out as the number of steps increases.

The properties of a random walk can be further analyzed using mathematical techniques such as probability theory and stochastic processes. Various extensions and modifications to the basic random walk model exist, including higher-dimensional random walks, biased random walks, and random walks with absorbing boundaries or obstacles.

Random walks find applications in many fields, including physics, finance, biology, and computer science. They provide a simple yet powerful framework for modeling and understanding complex systems with stochastic dynamics.