Towards a double-scaling limit for tensor models: probing sub-dominant orders

TL;DR

This work advances the understanding of double scaling in tensor models by computing and characterizing the next-to-leading order (NLO) contributions in the large-$N$ expansion of the iid $(D+1)$-colored tensor model. By identifying the NLO core graphs as a 2-dipole family and constructing the full nlo sector via melonic insertions and 1-dipole moves, the authors derive a closed relation for the connected 2-point function and extract the critical behavior, finding the same critical coupling $g_c$ as leading order but a distinct susceptibility exponent $\gamma_{\text{nlo}}=3/2$ (while $\gamma_{\text{lo}}=1/2$). This separation of exponents suggests a nontrivial double scaling limit, conjectured to occur when $N\to\infty$ and $g\to g_c$ with $N\left(1 - g/g_c\right)^{1/(D-2)}=\kappa$, under plausible dominance and resummability assumptions for the 2-dipole sector. The results provide a concrete analytic pathway toward richer continuum limits in tensor models and TGFTs, potentially illuminating spherical topologies and their gravitational interpretations via Regge/EDT formalisms.

Abstract

The definition of a double-scaling limit represents an important goal in the development of tensor models. We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the IID tensor models. We show that the radius of convergence of the NLO series coincides with that of the leading order melonic sector. Meanwhile, the value of the susceptibility exponent at NLO is 3/2, signaling a departure from the leading order behaviour. Both pieces of information provide clues for a non-trivial double-scaling limit, for which we put forward some precise conjecture.

Figures (15)

• Figure 4: The supermelon core graph, denoted $\mathcal{G}_{\textsc{supermelon}}$.
• Figure 5: An elementary melon of species $i_0$.
• Figure 7: The core graphs at nlo, denoted collectively by $\mathcal{G}_{\textsc{2--dipole}}$.
• Figure 8: A 2--dipole of species $\{i_0i_1\}$.
• Figure 9: The creation of a second 1--dipole, of species $\{i_0\}$, which is not equivalent to the insertion of an elementary melon. The edges contributing to the 1--dipole are tagged.

Theorems & Definitions (19)

• Proposition 3.1
• Definition 3.2
• Proposition 3.3
• Proposition 3.4
• Definition A.1
• Definition A.2
• Definition A.3
• Lemma A.4
• proof
• Definition A.5