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Using an $O(N)$-symmetric toy model QFT in zero space-time dimensions wediscuss several aspects and limitations of the $\frac{1}{N}$-expansion. Wedemonstrate, how slight modifications in a classical UV action can lead the$\frac{1}{N}$-expansion astray and how the infinite-$N$ limit may alterfundamental properties of a QFT. Thereby we present the problem of calculatingcorrelation functions from two totally different perspectives: First, weexplicitly analyze our model within an $\frac{1}{N}$-saddle-point expansion andshow its limitations. Secondly, we picture the same problem within theframework of the Functional Renormalization Group. Applying novel analogiesbetween (F)RG flow equations and numerical fluid dynamics from parts I and IIof this series of publications, we recast the calculation of expectation valuesof our toy model into solving a highly non-linear but exactadvection(-diffusion) equation. In doing so, we find that the applicability ofthe $\frac{1}{N}$-expansion to our toy model is linked to freezing shock wavesin 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 twoopposing shock waves in field space.

physical review. d/physical review. d.

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs  as numerical test cases. III. Shock and rarefaction waves in RG flows reveal  limitations of the $N\rightarrow\infty$ limit in $O(N)$-type models