Consider a grid with nodes (*i*, *j*) spaced one unit apart in both axes. These nodes, at this point, have nothing to do with electrostatics –– there are no voltages, charges, or fields present. Instead, there is a particle undergoing a *random walk*. Brownian motion,^{1} for example, is a random walk of particles. Random walks have various defining characteristics; the random walk discussed here is defined as follows:

- There is a particle sitting at one of the nodes. At time (
*m*− 1) the probability of the particle being at node (*i, j*) is*P*_{m−1}(*i*,*j*). - Every second the particle will quickly jump to an adjacent node.
*Adjacent*means to the left (−*x*), right (+*x*), up (+*y*) or down (−*y*). Diagonal jumps and multinode jumps are not allowed. - All of the jumps have equal probability; that is, the probability of any given jump is .

Four examples of such a walk, reminiscent of a drunkard wandering through a parking lot, are shown in Figure 11.1. In each of these examples the particle started at (0,0).

At a given time, a particle can arrive at node (*i*, *j*) only if it started from one of the (four) adjacent nodes. The probability of a particle arriving at ...

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