  Using an $O(N)$-symmetric toy model QFT in zero space-time dimensions we
discuss several aspects and limitations of the $\frac{1}{N}$-expansion. We
demonstrate, how slight modifications in a classical UV action can lead the
$\frac{1}{N}$-expansion astray and how the infinite-$N$ limit may alter
fundamental properties of a QFT. Thereby we present the problem of calculating
correlation functions from two totally different perspectives: First, we
explicitly analyze our model within an $\frac{1}{N}$-saddle-point expansion and
show its limitations. Secondly, we picture the same problem within the
framework of the Functional Renormalization Group. Applying novel analogies
between (F)RG flow equations and numerical fluid dynamics from parts I and II
of this series of publications, we recast the calculation of expectation values
of our toy model into solving a highly non-linear but exact
advection(-diffusion) equation. In doing so, we find that the applicability of
the $\frac{1}{N}$-expansion to our toy model is linked to freezing shock waves
in field space in the FRG-fluid dynamic picture, while the failure of the
$\frac{1}{N}$-expansion in this context is related to the annihilation of two
opposing shock waves in field space.
