  Rank-d Tensorial Group Field Theories are quantum field theories defined on a
group manifold $G^{\times d}$, which represent a non-local generalization of
standard QFT, and a candidate formalism for quantum gravity, since, when
endowed with appropriate data, they can be interpreted as defining a field
theoretic description of the fundamental building blocks of quantum spacetime.
Their renormalisation analysis is crucial both for establishing their
consistency as quantum field theories, and for studying the emergence of
continuum spacetime and geometry from them. In this paper, we study the
renormalisation group flow of two simple classes of TGFTs, defined for the
group $G=\mathbb{R}$ for arbitrary rank, both without and with gauge invariance
conditions, by means of functional renormalisation group techniques. The issue
of IR divergences is tackled by the definition of a proper thermodynamic limit
for TGFTs. We map the phase diagram of such models, in a simple truncation, and
identify both UV and IR fixed points of the RG flow. Encouragingly, for all the
models we study, we find evidence for the existence of a phase transition of
condensation type.
