Random Walk

TL;DR

• Models movement as a sequence of random, stepwise changes rather than a predictable trajectory.
• Key properties: each step is independent of prior steps (“no ‘memory’”) and the path tends to diffuse so distance from the start increases over time.
• Commonly used to represent unpredictable systems influenced by many variables, for example financial prices or a person’s path through a city.

Definition

A random walk is a mathematical concept that describes the movement of an object or a series of events as a sequence of random steps.

Explanation

A random walk represents a process where each successive step is determined by chance and not by the history of previous steps. Because each step is independent, the process has no “memory”: the next step does not depend on how the process arrived at its current state. Over time, a random walk tends to diffuse, meaning the distance traveled typically increases as the process continues. Random walks are useful for modeling systems influenced by many unpredictable factors, where outcomes cannot be predicted with certainty.

Examples

Stock price movement

Consider a company whose stock price starts at $50 per share. Over the course of a year the stock price may fluctuate due to earnings reports, market trends, management changes, and other factors. The price might increase or decrease by a few dollars on any given day, and the overall trend is unpredictable because the price is influenced by a large number of variables.

Person walking through a city

A person walking through a crowded city encounters obstacles such as other pedestrians, street signs, and buildings. They may walk in a straight line or change course to avoid obstacles or take a different route. Each step is influenced by immediate circumstances, so the person’s next step cannot be predicted with certainty; over time the person tends to travel further from their starting point despite local back-and-forth movements.

Use cases

• Modeling the behavior of financial markets and price movements.
• Helping plan routes or navigate environments where movement choices are influenced by many local factors.

Notes or pitfalls

• Outcomes are unpredictable because many variables influence each step.
• The “no ‘memory’” property means past steps do not determine future steps, so historical patterns do not guarantee future behavior.
• Over time the path typically diffuses, so deviations from the starting point generally increase.

Related terms

• no “memory” (step independence)
• diffusion