A frog sits on one of the 12 numbers on the face of a clock. Every minute the frog jumps left or right with equal probability.  If the frog starts as 12, then after one jump the frog is on 11 or 1, and so on. The frog-jumping process is an example of a random walk.  Successive positions of the frog (e.g., 12, 11, 10, 11, 12, 1, 2, 3, 2, . . . ) form a random sequence called a Markov chain, a class of mathematical objects that are enormously useful in modeling complex physical systems.

The Ising model is an important model in statistical physics used to explain ferromagnetism.  The system is made up of a large number of "sites" that take on one of two values (-1 or +1, white or black), with sites exerting some energy influence on their neighbors.  Below is a simulated configuration of the model in its "equilibrium" state.  After introducing Markov chains, we'll show how frog jumping, and random walks, were used to draw the picture.