Random Walk--2-Dimensional

In a plane, consider a sum of two-dimensional vectors with random orientations. Use phasor notation, and let the phase of each vector be random. Assume unit steps are taken in an arbitrary direction (i.e., with the angle uniformly distributed in and not on a lattice), as illustrated above. The position in the complex plane after steps is then given by

(1)

which has absolute square

			(2)
			(3)
			(4)

Therefore,

(5)

Each unit step is equally likely to be in any direction ( and ). The displacements are random variables with identical means of zero, and their difference is also a random variable. Averaging over this distribution, which has equally likely positive and negative values yields an expectation value of 0, so

(6)

The root-mean-square distance after unit steps is therefore

(7)

so with a step size of , this becomes

(8)

In order to travel a distance ,

(9)

steps are therefore required.

Amazingly, it has been proven that on a two-dimensional lattice, a random walk has unity probability of reaching any point (including the starting point) as the number of steps approaches infinity.

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