The Large D Limit of Planar Diagrams

TL;DR

The paper establishes a nontrivial large-D limit for O(D) invariant matrix models where, after first taking N→∞, the sum over planar diagrams collapses to a tractable melonic subset. By introducing a scaling λ_B = D^{-g(B)} t_B, planar diagrams acquire a well-defined 1/√D expansion, with the leading generalized melon graphs summable via closed Schwinger-Dyson equations. The complex-matrix case can be fully analyzed, while the Hermitian case requires orienting planar diagrams to preserve a bipartite structure, which can be achieved for planar graphs. This framework connects matrix quantum mechanics to SYK/tensor-model techniques and opens avenues for non-perturbative studies of black-hole-like dynamics and potential generalizations to matrix-tensor systems.

Abstract

We show that in $\text{O}(D)$ invariant matrix theories containing a large number $D$ of complex or Hermitian matrices, one can define a $D\rightarrow\infty$ limit for which the sum over planar diagrams truncates to a tractable, yet non-trivial, sum over melon diagrams. In particular, results obtained recently in SYK and tensor models can be generalized to traditional, string-inspired matrix quantum mechanical models of black holes.

Figures (5)

• Figure 1: From left to right, colored, ribbon and stranded graphs associated to the quartic interaction terms $\mathop{\rm tr}\nolimits (X_{\mu}X_{\mu}^{\dagger}X_{\nu}X_{\nu}^{\dagger})$ (up) and $\mathop{\rm tr}\nolimits (X_{\mu}X_{\nu}^{\dagger}X_{\mu}X_{\nu}^{\dagger})$ (down). The r-graphs have genus zero and one-half respectively. We have chosen to orient the ribbons in the stranded graph, corresponding to the case of complex matrices.
• Figure 2: The basic two-loop diagram from which melons are built in the $\lambda\sqrt{D}\mathop{\rm tr}\nolimits (X_{\mu}X_{\nu}X_{\mu}X_{\nu})$ model (upper-left), its stylized representation (lower left) and a typical stylized melon diagram (right). All the diagrams contributing at leading order $N^{2}D$ are of this type.
• Figure 3: A planar four-loop Feynman diagram of order $N^{2}D^{-1}$ in the Hermitian matrix model in the s-graph (left) and c-graph (right) representations. The diagram contains two vertices $\mathop{\rm tr}\nolimits (X_{\mu}X_{\mu}X_{\nu}X_{\nu})$ and $\mathop{\rm tr}\nolimits (X_{\mu}X_{\nu}X_{\rho}X_{\nu}X_{\mu}X_{\rho})$ of genera zero and one respectively. The number of $\text{U}(N)$ faces $f=5$ does not match $F_{\text{vg}}+F_{\text{vr}}=3$.
• Figure 4: Another c-graph representation (left) of the Feynman graph depicted on Fig. \ref{['fig3']}, for which the violet lines respect the bipartite structure and thus for which $F_{\text{vg}}+F_{\text{vr}}=f=5$. Equivalently, one can orient the propagators of the Hermitian model s-graph to make it indistinguishable from a complex model graph (right).
• Figure 5: Steps from a stranded planar Feynman graph for the Hermitian matrix model to a c-graph whose violet lines respect the bipartite structure.