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

Automated summary

This paper systematically studies quantum field theory on random trees, establishing the general power counting of averaged Feynman amplitudes and demonstrating their behavior in an effective space of dimension 4/3. In the renormalizable case, convergence of averaged amplitudes for completely convergent graphs is proven, laying the groundwork for perturbative renormalization. Possible implications for an SYK-like model on random trees are briefly discussed.