info prev up next book cdrom email home

Random Walk--2-D

\begin{figure}\begin{center}\BoxedEPSF{RandomWalk2D.epsf scaled 1200}\end{center}\end{figure}

In a Plane, consider a sum of $N$ 2-D Vectors with random orientations. Use Phasor notation, and let the phase of each Vector be Random. Assume $N$ unit steps are taken in an arbitrary direction (i.e., with the angle $\theta$ uniformly distributed in $[0,2\pi)$ and not on a Lattice), as illustrated above. The position $z$ in the Complex Plane after $N$ steps is then given by

z = \sum_{j=1}^N e^{i\theta_j},
\end{displaymath} (1)

which has Absolute Square
$\displaystyle \vert z\vert^2$ $\textstyle =$ $\displaystyle \sum_{j=1}^N e^{i\theta_j} \sum_{k=1}^N e^{-i\theta_k} = \sum_{j=1}^N \sum_{k=1}^N e^{i(\theta_j-\theta_k)}$  
  $\textstyle =$ $\displaystyle N + \sum_{\scriptstyle j,k=1\atop\scriptstyle k\not=j}^N e^{i(\theta_j-\theta_k)}.$ (2)

\left\langle{\vert z\vert^2}\right\rangle{} = N + \left\lang...
...riptstyle k\not=j}^N e^{i(\theta_j-\theta_k)}}\right\rangle{}.
\end{displaymath} (3)

Each step is likely to be in any direction, so both $\theta_j$ and $\theta_k$ 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
\left\langle{\vert z\vert^2}\right\rangle{} = N.
\end{displaymath} (4)

The root-mean-square distance after $N$ unit steps is therefore
\vert z\vert _{\rm rms} = \sqrt{N},
\end{displaymath} (5)

so with a step size of $l$, this becomes
d_{\rm rms} = l\sqrt{N}.
\end{displaymath} (6)

In order to travel a distance $d$
N \approx \left({d\over l}\right)^2
\end{displaymath} (7)

steps are therefore required.


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

info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein