Fixed-step isotropic random flight

A fixed-step isotropic random flight is a random walk in continuous space with independent steps of a common length $\ell$, each direction uniformly distributed over all directions. In $d$ dimensions, write its position as $S_n=\ell {\sum }_{j=1}^n{U}_j$, where the $U_j$ are independent unit vectors. Rotational symmetry gives $\mathbb{E}{U}_j=0$ and $\mathbb{E}(U_j{U}_j^{\mathsf{T}})=I_d/d$. Independence then givesThus 100 steps of length one have total path length 100 but root mean square displacement 10. A zero mean vector does not mean that a typical walker remains at its starting point.^[ref]

An exact two-step example in three dimensions shows how the endpoint spreads. The cosine $C=U_1\cdot U_2$ is uniform on $[-1,1]$, and $R^2=\Vert S_2{\Vert }^2=2{\ell }^2(1+C)$. Changing variables gives the radial probability density functionThis is a density per unit distance; a density per unit volume includes the spherical volume factor $4\pi r^2$. Uniform polar angle would give a different model: isotropic simulation in three dimensions samples the cosine of that angle uniformly.

In three dimensions the one-step characteristic function is $\sin (\ell k)/(\ell k)$, with value one at $k=0$, so the $n$-step characteristic function is its $n$th power. For a step duration $\tau$, the diffusion limit keeps $D={\ell }^2/(6\tau )$ fixed as $\ell ,\tau$ tend to zero. It yields the Gaussian diffusion equation kernel. At finite $n$, however, the walker cannot travel farther than $n\ell$; a Gaussian distribution is a limiting approximation, not the exact finite-step law.^[ref]