Random Walk--3-D

$\begin{figure}\begin{center}\BoxedEPSF{RandomWalk3DLattice.epsf}\end{center}\end{figure}$

On a 3-D Lattice, a random walk has less than unity probability of reaching any point (including the starting point) as the number of steps approaches infinity. The probability of reaching the starting point again is 0.3405373296.... This is one of Pólya's Random Walk Constants.

See also Pólya's Random Walk Constants, Random Walk--1-D, Random Walk--2-D