Perturbative Quantum Field Theory on Random Trees

In this paper we start a systematic study of quantum field theory on random trees. Using precise probability estimates their Galton–Watson branches and multiscale analysis, establish the general power counting averaged Feynman amplitudes check that they behave indeed as living an effective space dimension 4/3, spectral “just renormalizable” case prove convergence amplitude any completely convergent graph, basic localization subtraction required for perturbative renormalization. Possible consequences SYK-like model trees are briefly discussed.