  The perturbative expansion of tensorial field theories in Feynman graphs can
be interpreted as weighted generating series of some piecewise linear
varieties. This simple fact establishes a link between two a priori distinct
fields: the combinatorics of discrete manifolds on one hand and tensorial field
theories on the other hand. In this thesis, we study different aspects
revolving around this connection between combinatorics and field theory. First,
we consider constellations model, which generalize maps and their algebraic
properties. This makes them suited to probe the b-deformation, a deformation of
the algebra of symmetric functions. We will study the constraints satisfied by
the generating series of cubical b-deformed constellations. Second, we analyze
the double scaling limit of particular tensor models of order 3. For tensor of
order greater than two, the nature of the 1/N-expansion is qualitatively
different from the matrix case of order 2. In particular, only the leading
order graphs are fully characterized. Despite this fact, it is possible to
identify graphs of subleading orders contributing to the double scaling limit
by implementing the scheme decomposition for Feynman graphs of these theories.
An analysis of the singularity of the schemes then allows us to give a complete
characterization of the graphs contributing to the double scaling limit.
Finally, we investigate a particular link between a tensor and a vector field
theory which both admit a melonic limit. Namely, we will show that we can
obtain the vectorial Amit-Roginski model by considering perturbations around a
classical solution of the Boulatov model, a tensorial theory. We give
sufficient conditions on the classical solution so that the effective action
for the perturbation around this solution takes the form of the Amit-Roginski
action.
