Erich Prisner

Random Walks 2

Now try a two-dimensional random walk. The person starts in the upper left corner and has to reach the lower right corner. A cell is adjacent to the cells to the left, the right, above, below, and in each stap the person goes to an adjacent cell, each neighbor having equal probability.

 Time:
 Time:

These and other grids, like 4*4 grids or on 5*5 grids could also be analyzed using recurrence relations. However things become more complex. What can be analyzed is the probability for the person to go from start to target in the smallest possible time. It is the first nonzero entry in the right probability distribution applet, but the value can also be computed for this and any graph.

Another interesting feature can be seen by clicking the "ordinary step" button repeatedly. That means that the person has no target and goes on and on, even after meeting the lower right cell. Of course, the probabilities in the cells alternate between 0 and positive values, but interestingly after some time a pattern in the probabilities can be seen. The inner cells are more likely than the border cells, and the corner cells are most unlikely.

Erich Prisner 2004