Random Walk--2-D

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

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

which has Absolute Square

$$\vert z\vert^2 = \sum_{j=1}^N e^{i\theta_j} \sum_{k=1}^N e^{-i\theta_k} = \sum_{j=1}^N \sum_{k=1}^N e^{i(\theta_j-\theta_k)}$$

$$= N + \sum_{\scriptstyle j,k=1\atop\scriptstyle k\not=j}^N e^{i(\theta_j-\theta_k)}.$$ (2)

Therefore,

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

$$\left\langle{\vert z\vert^2}\right\rangle{} = N.$$ (4)

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

$$\vert z\vert _{\rm rms} = \sqrt{N},$$ (5)

so with a step size of $l$, this becomes

$$d_{\rm rms} = l\sqrt{N}.$$ (6)

In order to travel a distance $d$

$$N \approx \left({d\over l}\right)^2$$ (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