Melons are branched polymers

TL;DR

This paper demonstrates that melonic graphs, prominent in tensor models, are exactly equivalent to branched polymers, characterized by specific Hausdorff and spectral dimensions, thus clarifying their geometric nature.

Contribution

The paper establishes that melonic graphs are precisely branched polymers, providing exact dimensions and confirming their geometric structure in tensor models.

Findings

Melonic graphs have Hausdorff dimension 2.

Melonic graphs have spectral dimension 4/3.

Melonic graphs are equivalent to branched polymers.

Abstract

Melonic graphs constitute the family of graphs arising at leading order in the 1/N expansion of tensor models. They were shown to lead to a continuum phase, reminiscent of branched polymers. We show here that they are in fact precisely branched polymers, that is, they possess Hausdorff dimension 2 and spectral dimension 4/3.