Melons are branched polymers

TL;DR

This work shows that melonic graphs, which dominate the leading $1/N$ expansion in tensor models, are in fact branched polymers by establishing identical Hausdorff and spectral dimensions, $d_H=2$ and $d_S=4/3$. The authors leverage bijections between rooted melonic graphs and colored $(D+1)$-ary trees, melonic $D$-balls, and stack spheres, then analyze geometry through depth, distance, and sub-word structure. A central result is the Gromov–Hausdorff convergence of melonic $D$-balls to the continuum random tree under a nontrivial rescaling by $\Lambda_{\Delta}\sqrt{(D+1)n/D}$, which yields $d_H=2$, while diffusion analysis on rooted melonic graphs yields $d_S=4/3$. The combination of these results identifies melonic graphs with branched polymers and provides a rigorous geometric understanding of their continuum limit, with implications for the interpretation of tensor-model universality and gravity-inspired random geometries.

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.

Figures (6)

• Figure 4: The unique stack $D$--sphere at $p=1$ (for $D=2$).
• Figure 5: Coloring the vertices of the elementary stack sphere (for $D=2$). Only the active 2--simplices are drawn.
• Figure 7: The words associated to some vertices of a colored rooted $(D+1)$--ary tree.
• Figure 9: A rooted melonic graph $\mathcal{M}$ with sub--melons $\mathcal{M}^i$.
• Figure 10: A closed $D$-colored graph and its associated trace invariant (for $D=3$).

Theorems & Definitions (7)

• Lemma 1
• Lemma 2
• Theorem 1
• Lemma 3
• Lemma 4
• Lemma
• Theorem