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Perturbative Quantum Field Theory on Random Trees

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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.

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Notes

  1. The usual spine obtained by conditioning a critical Galton–Watson tree by non-extinction corresponds to a one dimensional half-space. However it should be straightforward to symmetrize the spine to get a full one dimensional space.

  2. See however e.g. [30, 31] for statistical mechanic models on random trees and graphs.

  3. This proper time is nothing but Feynman’s parameter in high energy physics language.

  4. The graph distance d(x, y) denotes the smallest number of steps on the tree needed to connect x to y.

  5. For the upper inequality, we used that for any \(m>0\),  \( 3/(3+m^2) > 1/(1+m^2)\). The lower one is obtained by comparing the Taylor expansions of both members around \(m=0\). K is chosen such that the inequality between the rational function and the exponential holds. Whereas c is independent of m, if \(m<1\), \(K>5\) is enough.

  6. In a usual theory there is no \(x_0\) dependence because of translation invariance, but for a particular tree T there is no such invariance.

  7. We refer to Ch. 2 of [1] for details on going from the discrete to continuous time propagators, the exponential factor stemming from the mass regulator.

  8. We do not try to make \(\beta \) optimal. We expect that a tighter probabilistic analysis could prove subfactorial growth in n for \({\mathbb {E}}( A_G)\).

  9. The attentive reader wondering about the factor 54 will find that it comes from the fact that \((N-4)/3\ge N/9\) for \(N\ge 6\) and that there are 6 different pairs at a \(\phi ^4\) vertex.

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Acknowledgements

We thank the organizers of the workshop “Higher spins and holography” for their invitation and the Erwin Schrödinger Institute for the stimulating scientific atmosphere provided during this workshop where part of this work was elaborated. N.D. would like to thank J. Ben Geloun and T. El Khaouja for useful comments.

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  1. Laboratoire de physique théorique, CNRS UMR6827, Université Paris-Sud, Université Paris-Saclay, 91405, Orsay, France

    Nicolas Delporte & Vincent Rivasseau

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  1. Nicolas Delporte
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Correspondence to Nicolas Delporte.

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Delporte, N., Rivasseau, V. Perturbative Quantum Field Theory on Random Trees. Commun. Math. Phys. 381, 857–887 (2021). https://doi.org/10.1007/s00220-020-03874-2

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