Double scaling limit of multi-matrix models at large $D$

TL;DR

This paper investigates the double scaling limit of two multi-matrix models with large N and D, revealing a shared universality class for the most singular schemes, extending combinatorial insights to quantum field theory.

Contribution

It introduces a combinatorial framework for analyzing double scaling limits in multi-matrix models, connecting large D expansions with scheme classifications and universality classes.

Findings

Most singular schemes match those in previous tetrahedral models

Double scaling limit at large D is summable in these models

Identifies a new universality class distinct from 1-matrix models

Abstract

In this paper, we study a double scaling limit of two multi-matrix models: the $U(N)^2 \times O(D)$-invariant model with all quartic interactions and the bipartite $U(N) \times O(D)$-invariant model with tetrahedral interaction ($D$ being here the number of matrices and $N$ being the size of each matrix). Those models admit a double, large $N$ and large $D$ expansion. While $N$ tracks the genus of the Feynman graphs, $D$ tracks another quantity called the grade. In both models, we rewrite the sum over Feynman graphs at fixed genus and grade as a finite sum over combinatorial objects called schemes. This is a result of combinatorial nature which remains true in the quantum mechanical setting and in quantum field theory. Then we proceed to the double scaling limit at large $D$, i.e. for vanishing grade. In particular, we find that the most singular schemes, in both models, are the same as those found in Benedetti et al. for the $U(N)^2 \times O(D)$-invariant model restricted to its tetrahedral interaction. This is a different universality class than in the 1-matrix model whose double scaling is not summable.