In a Plane, consider a sum of 2-D 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) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
Amazingly, it has been proven that on a 2-D Lattice, a random walk has unity probability of reaching any point (including the starting point) as the number of steps approaches Infinity.
See also Pólya's Random Walk Constants, Random Walk--1-D, Random Walk--3-D
© 1996-9 Eric W. Weisstein