More on the New Large $D$ Limit of Matrix Models

TL;DR

This work extends the recently proposed large $D$ limit for matrix models to include arbitrary multi-trace interactions and general correlation functions across complex and Hermitian matrices with distinct $ ext{U}(N)$ and $ ext{O}(D)$ symmetries. It develops a detailed double expansion in $N$ and $D$, proving nonnegativity of the exponents ($h$, $\ell$) that control the powers of $N$ and $D$, and characterizes leading diagrams as generalized melons with planar constraints; higher-genus contributions require a tracelessness condition in many cases. The paper also analyzes correlation functions, establishing scaling rules and factorization, and discusses model-building implications including unstable vs stable bosonic theories and linearly realized supersymmetry, showing compatibility of the new scaling with SUSY. Together, these results provide a comprehensive, tractable framework for solvable large-$N$/$D$ matrix theories with rich interaction structures, relevant for holography, chaos, and beyond.

Abstract

In this paper, we extend the recent analysis of the new large $D$ limit of matrix models to the cases where the action contains arbitrary multi-trace interaction terms as well as to arbitrary correlation functions. We discuss both the cases of complex and Hermitian matrices, with $\text{U}(N)^{2}\times\text{O}(D)$ and $\text{U}(N)\times\text{O}(D)$ symmetries respectively. In the latter case, the new large $D$ limit is consistent for planar diagrams; at higher genera, it crucially requires the tracelessness condition. For similar reasons, the large $N$ limit of tensor models with reduced symmetries is typically inconsistent already at leading order without the tracelessness condition. We also further discuss some interesting properties of purely bosonic models pointed out recently and explain that the standard argument predicting a non-trivial IR behaviour in fermionic models à la SYK does not work for bosonic models. Finally, we explain that the new large $D$ scaling is consistent with linearly realized supersymmetry.

Figures (10)

• Figure 1: Fat graph and colored graph for the interaction vertex $\mathop{\rm tr}\nolimits X_{\mu}X_\nu^\dagger \,\mathop{\rm tr}\nolimits X_\mu X_\rho^\dagger X_\nu X_\rho^\dagger\,\mathop{\rm tr}\nolimits X_\sigma X_\sigma^\dagger$, with $c=2$, $t=3$ and $g=1/2$.
• Figure 2: Melonic moves for the $\mathop{\rm tr}\nolimits X_{\mu}X_{\nu}^{\dagger} X_{\mu}X_{\nu}^{\dagger}$ and $\mathop{\rm tr}\nolimits X_{\mu}X_{\nu}^{\dagger}X_{\rho} X_{\mu}^{\dagger}X_{\nu}X_{\rho}^{\dagger}$ interactions.
• Figure 3: Moves increasing the genus by one unit at fixed $\ell$ for the $\mathop{\rm tr}\nolimits X_{\mu}X_{\nu}^{\dagger} X_{\mu}X_{\nu}^{\dagger}$ and $\mathop{\rm tr}\nolimits X_{\mu}X_{\nu}^{\dagger}X_{\rho} X_{\mu}^{\dagger}X_{\nu}X_{\rho}^{\dagger}$ interactions.
• Figure 4: Stranded graph and colored graph for the interaction vertex $\mathop{\rm tr}\nolimits X_{\rho}X_{\rho}^{\dagger}\,\mathop{\rm tr}\nolimits X_{\mu} X_{\nu}^{\dagger} X_{\mu} X_{\nu}^{\dagger}$ discussed in the main text, with $c=2$, $t=2$ and $g=1/2$.
• Figure 5: Structure of the leading order vacuum graphs for the multi-trace model with interaction vertex $\mathop{\rm tr}\nolimits X_{\rho}X_{\rho}^{\dagger}\,\mathop{\rm tr}\nolimits X_{\mu} X_{\nu}^{\dagger} X_{\mu} X_{\nu}^{\dagger}$. $G$ is the leading two-point function.