# Perturbative Quantum Field Theory on Random Trees

## TL;DR

This paper develops a systematic framework for quantum field theory on random trees, revealing that their effective spectral dimension is 4/3 and establishing foundational renormalization results.

## Contribution

It introduces a multiscale analysis and power counting for quantum fields on random trees, demonstrating their behavior as a 4/3-dimensional space and proving convergence in the renormalizable case.

## Key findings

- Effective spectral dimension of 4/3 for random trees
- Convergence of averaged amplitudes in the renormalizable case
- Foundational localization and subtraction estimates for renormalization

## Abstract

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.