  In the tensorial group field theory approach to quantum gravity, the theory
is based on discrete building blocks and continuum spacetime is expected to
emerge from their collective dynamics, possibly at criticality, via a phase
transition. On a compact group of fixed volume this can be expected to be only
possible in a large-volume or thermodynamic limit. Here we show how phase
transitions are possible in TGFTs in two cases: a) considering the non-local
group degrees of freedom on a non-compact Lie group instead of a compact one
(or taking a large-volume limit of a compact group); b) in models including
$\mathbb{R}$-valued local degrees of freedom (that can be interpreted as
discrete scalar fields, often used in this context to provide a matter
reference frame). After adapting the Landau-Ginzburg approach to this setting
of mixed local/non-local degrees of freedom, we determine the critical
dimension beyond which there is a Gaussian fixed point and a continuous phase
transition which can be described by mean-field theory. This is an important
step towards the realization of a phase transition to continuum spacetime in
realistic TGFT models for quantum gravity.
