Random Walks

There exists several simple models of random walks. It can be either fixed stepsize but random directions or varying stepsize.

Fixed Stepsize

A simple model is that we assume each step is a step of size \epsilon, but with random directions. The position of this random walker at step N is the summation of all the steps (vectors),

\vec X = \sum_i^N x_i,

where x_i is the vector that represents step i.

What we want to find out is the places that the random walker explored after N steps. The corresponding quantity that represents it is \sqrt{\langle \vec X^2 \rangle}.

From the idea of random walk, we know that

\langle \vec x_i \rangle =& 0 \\
\langle \vec x_i \cdot \vec x_j \rangle =& 0.

Thus

\langle \vec X^2 \rangle = & \sum_i \vec x_i\cdot \vec x_i \\
=& N \epsilon^2.

Then we find out that

(1)\bar X = \sqrt{\langle \vec X^2 \rangle} = \sqrt{N}\epsilon.

Significance of Dimension

From the root-mean-squared distance Eq. (1) we can define the density of points. Suppose we have a continues version of this random walk. After time t, the random walker walked a distance vt. Meanwhile the random walker explored a region of radius \sqrt{t}v, which corresponds to a volume V \propto \sqrt{t}v. The density of walked points is defined as

\rho \propto \frac{t}{\sqrt{t}^d} = t^{1-d/2},

where d is the dimension of the space.

We spot this critical dimension d=2.

  1. d<2: the density of points at t\to\infty becomes \rho\to \infty. This called recurrent behavior. We are sure that after infinite time, we are going back to a point that we visited before.
  2. d=2: the density of points at t\to\infty becomes \rho\to \mathrm{Constant}. This derivation is wrong about this critical case. It should be \rho\to \ln t
  3. d>2: the density of points at t\to\infty becomes \rho\to 0. This is called transit behavior. We are not sure that we could go back to a point that we visited before.

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