Random walk

Authors: Alex Tee Neng Heng, David G. Green

A random walk is a process consisting of a sequence of discrete steps of fixed length, where the direction of each step is random and does not depend on the previous steps. Random walks are an abstraction for a range of processes observed in all sorts of natural complex systems. For instance, random Brownian motion of molecules in liquids caused by thermal energy represents a random walk, as well as diffusing gas molecules. In biology, random walks can be observed in the foraging behaviour or insects. The mathematical properties of random walks vary greatly depending on the dimensionality of the space in which the walk is undertaken. Some of such properties bare quite some fascinating surprises.

This demonstration plots the area visited by a number of agents engaged in random walks, starting from the centre of the field. Typically, there is a circular area surrounding the starting point within which every point has been visited (the size depends on the number of agents).

How to use the simulation

You need to have Java version 1.5 installed for your browser in order to run the simulation applet.

  • Set the  Number of walkers or agents participating in the random walk and the Number of iterations (steps) to simulate using the control panel at the top of the simulation window.
  • Set the probability for the change of direction. The probability can range between 0 and 1. When 0, the walk is not random and the agents will only move straight forward. If the probability is 1, the agents will choose a new, random direction at each time step. The values between 0 and 1 determine, how likely is an agent to change direction at each time step is.
  • Use the Start and the Reset buttons to control the flow of the simulation.


  • Try setting the number of iterations to a very large number and watch how the agents slowly cover the compete area.
  • Vary the number of agents and see how the shape of the area covered after a certain number of steps changes. Try 10 agents, 1 agent, 100 agents. What is different?
  • Try altering the probability for change of direction. How does is affect the shape of the area covered? And what about the size of the area?

Links and References

Demo screenshot