  The transport of non-thermal particles across a large-scale magnetic field in
the presence of magnetised turbulence has been a long-standing issue in
high-energy astrophysics. Of particular interest is the dependence of the
parallel and perpendicular mean free paths $\lambda_{\parallel}$ and
$\lambda_{\perp}$ on rigidity $\mathcal{R}$. We have revisited this important
issue with a view to applications from the transport of Galactic cosmic rays to
acceleration at astrophysical shocks. We have run test particle simulations of
cosmic ray transport in synthetic, isotropic Kolmogorov turbulence at
unprecedentedly low reduced rigidites $r_g/L_c \simeq 10^{-4}$, corresponding
to $\mathcal{R} \simeq 10 \, \text{TV}$ for a turbulent magnetic field of
$B_{rms} = 4 \, \mu\text{G}$ and correlation length $L_c = 30 \, \text{pc}$.
Extracting the (asymptotic) parallel and perpendicular mean free paths
$\lambda_{\parallel}$ and $\lambda_{\perp}$, we have found $\lambda_{\parallel}
\propto (r_g/L_c)^{1/3}$ as expected for a Kolmogorov turbulence spectrum. In
contrast, $\lambda_{\perp}$ has a faster dependence on $r_g/L_c$ for $10^{-2}
\lesssim r_g/L_c \lesssim 1$, but for $r_g/L_c \ll 10^{-2}$, also
$\lambda_{\perp} \propto (r_g/L_c)^{1/3}$. Our results have important
implications for the transport of Galactic cosmic rays.
