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.
