Random Walks

Suppose our random walks represent motion on a lattice, which means we only look at integer valued coordinates.

We will examine a 2-dimensional lattice, because it is easier to visualize, however a random walk can exist in any dimension. The walk will traverse only the lattice points by moving to a neighboring lattice point. The probability of moving to any neighboring lattice point is equal for all neighboring points, hence it is 1/2d ( where d is the dimension ). Therefore in 2-dimensional space the probability is 1/4 of visiting either of the four neighboring point. This is the basis for a simple random walk. There are extensions into more sophisticated walks, such as loop erased walks, or self-avoiding walks. 

Results: A random walks is they are recurrent in dimensions 1 and 2, while transient in dimensions bigger than 3. This means a random walk in 1 or two dimensional space will visit every point infinitely often. However a walk in dimension 3 or more has positive probability of never going back to any point.

Gambling with two players is connected to a 1 dimensional random walk, either you win ( move right ) or the house wins ( move left )

This walk is recurrent, hence we will eventually win as much money as we can imagine, and we will lose just as much. Unfortunately, one has to stop playing once we run out of money. The casino's have essentially unlimited money ( since they produce their own chips ). This means the casino can essentially play forever, but gamblers have limited funds. Therefore, random walks imply gamblers are at a disadvantage and must find other disequilibrium's to find a way to get ahead of the casino.