Melonic phase transition in group field theory

Understand

Group field theories have recently been shown to admit a 1/N expansion dominated by so-called `melonic graphs', dual to triangulated spheres.

• In this note, we deepen the analysis of this melonic sector.
• We obtain a combinatorial formula for the melonic amplitudes in terms of a graph polynomial related to a higher dimensional generalization of the Kirchhoff tree-matrix theorem.
• Simple bounds on these amplitudes show the existence of a phase transition driven by melonic interaction processes.