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Random Walk--2-Dimensional


RandomWalk2D

In a plane, consider a sum of N two-dimensional 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 [0,2pi) 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)^Ne^(itheta_j),
(1)

which has absolute square

|z|^2=sum_(j=1)^(N)e^(itheta_j)sum_(k=1)^(N)e^(-itheta_k)
(2)
=sum_(j=1)^(N)sum_(k=1)^(N)e^(i(theta_j-theta_k))
(3)
=N+sum_(j,k=1; k!=j)^(N)e^(i(theta_j-theta_k)).
(4)

Therefore,

 <|z|^2>=N+<sum_(j,k=1; k!=j)^Ne^(i(theta_j-theta_k))>.
(5)

Each unit step is equally likely to be in any direction (theta_j and theta_k). 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

 <|z|^2>=N.
(6)

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

 |z|_(rms)=sqrt(N),
(7)

so with a step size of l, this becomes

 d_(rms)=lsqrt(N).
(8)

In order to travel a distance d,

 N approx (d/l)^2
(9)

steps are therefore required.

RandomWalk2DLattice

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.


See also

Pólya's Random Walk Constants, Random Walk--1-Dimensional, Random Walk--3-Dimensional

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References

McCrea, W. H. and Whipple, F. J. W. "Random Paths in Two and Three Dimensions." Proc. Roy. Soc. Edinburgh 60, 281-298, 1940.

Cite this as:

Weisstein, Eric W. "Random Walk--2-Dimensional." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RandomWalk2-Dimensional.html

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