Critical behavior of colored tensor models in the large N limit

TL;DR

This work analyzes the leading-order structure of colored tensor models in the large-$N$ limit, showing that melonic (D-bubble with two vertices) triangulations dominate and can be mapped bijectively to decorated $(D+1)$-ary trees. The authors derive a self-consistent melonic equation, solve it exactly, and demonstrate that the melonic sector is summable with a universal critical exponent $eta = 1/2$, signaling a continuum limit dominated by branched polymers. They establish that the dominant DT phase has $d_H=2$ and provide detailed large-volume scaling, including a continuum-limit recipe via $g o g_c$ and lattice spacing $a o 0$, with a renormalized cosmological parameter $ ilde{\Lambda}_{ m R}$. The results suggest a universal, dimension-independent critical behavior and connect dynamical triangulations in higher dimensions to branched polymer physics, offering a tractable route to understand continuum limits in tensor models and potential universality across model variants.

Abstract

Colored tensor models have been recently shown to admit a large N expansion, whose leading order encodes a sum over a class of colored triangulations of the D-sphere. The present paper investigates in details this leading order. We show that the relevant triangulations proliferate like a species of colored trees. The leading order is therefore summable and exhibits a critical behavior, independent of the dimension. A continuum limit is reached by tuning the coupling constant to its critical value while inserting an infinite number of pairs of D-simplices glued together in a specific way. We argue that the dominant triangulations are branched polymers.

Figures (8)

• Figure 1:
• Figure 2: The jacket ${\cal J}$.
• Figure 3:
• Figure 4: First order.
• Figure 5: Second order.

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Proposition 1
• Proposition 2
• Lemma 1
• Proposition 3
• Proposition 4
• Remark 1
• Proposition 5