On the large $D$ expansion of Hermitian multi-matrix models

Abstract

We investigate the existence and properties of a double asymptotic expansion in $1/N^2$ and $1/\sqrt{D}$ in $\mathrm{U}(N)\times \mathrm{O}(D)$ invariant Hermitian multi-matrix models, where the $N\times N$ matrices transform in the vector representation of $\mathrm{O}(D)$. The crucial point is to prove the existence of an upper bound $\eta (h)$ on the maximum power $D^{1+\eta (h)}$ of $D$ that can appear for the contribution at a given order $N^{2-2h}$ in the large $N$ expansion. We conjecture that $\eta (h)=h$ in a large class of models. In the case of traceless Hermitian matrices with the quartic tetrahedral interaction, we are able to prove that $\eta (h)\le 2h$; the sharper bound $\eta (h)=h$ is proven for a complex bipartite version of the model, with no need to impose a tracelessness condition. We also prove that $\eta (h)=h$ for the Hermitian model with the sextic wheel interaction, again with no need to impose a tracelessness condition.

28 pages, 20 figures; v2: refs added, matches published version

References in corpus (26)

• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• 2D Gravity and Random Matrices
• Quantum Field Theory in the Large N Limit: a review
• Holographic metals and the fractionalized Fermi liquid
• Critical behavior of colored tensor models in the large N limit
• Uncolored Random Tensors, Melon Diagrams, and the SYK Models
• Colored Tensor Models - a Review
• The complete 1/N expansion of colored tensor models in arbitrary dimension
• Non-Fermi liquid regime of a doped Mott insulator
• Random tensor models in the large N limit: Uncoloring the colored tensor models
• AdS${}_2$ holography and the SYK model
• The 1/N expansion of colored tensor models in arbitrary dimension
• The 1/N expansion of colored tensor models
• Mean-Field Theory of a Quantum Heisenberg Spin Glass
• $O(N)$ Random Tensor Models
• New 1/N expansions in random tensor models
• The 1/N expansion of multi-orientable random tensor models
• The Multi-Orientable Random Tensor Model, a Review
• The $1/N$ expansion of the symmetric traceless and the antisymmetric tensor models in rank three
• On Large $N$ Limit of Symmetric Traceless Tensor Models
• The $1/N$ expansion of tensor models with two symmetric tensors
• SYK-like tensor quantum mechanics with $\mathrm{Sp}(N)$ symmetry
• Large $N$ Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension $d\ge 2$
• More on the New Large $D$ Limit of Matrix Models
• Large $N$ limit of irreducible tensor models: $O(N)$ rank-$3$ tensors with mixed permutation symmetry
• Melonic Dominance in Subchromatic Sextic Tensor Models

Cited by in corpus (4)

• The Large D Limit of Einstein's Equations
• Double scaling limit of multi-matrix models at large $D$
• Double scaling limit for the $O(N{)}^3$-invariant tensor model
• Melonic large $N$ limit of $5$-index irreducible random tensors