= 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 $\mathbb E(U_jU_j^{\mathsf T})=I_d/d$. Independence then gives
$$
\mathbb E S_n=0,\qquad \operatorname{Cov}(S_n)=\frac{n\ell^2}{d}I_d,\qquad \boxed{\mathbb E\lVert S_n\rVert^2=n\ell^2.}
$$
Thus 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.<Physical Applications of Stochastic Processes by V. Balakrishnan>[Lecture 4, section 2.11, printed pages 35–36: model, independent steps and mean squared displacement]{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 $R^2=\lVert S_2\rVert^2=2\ell^2(1+C)$. Changing variables gives the radial <probability density function>
$$
f_R(r)=\frac{r}{2\ell^2}\quad(0\le r\le2\ell),\qquad \mathbb E R=\frac{4\ell}{3}.
$$
This 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.<Physical Applications of Stochastic Processes by V. Balakrishnan>[Section 2.11, printed pages 37–39: Fourier transform and diffusion coefficient]{ref}
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