A ribbon graph derivation of the algebra of functional renormalization for random multi-matrices with multi-trace interactions

We focus on functional renormalization for ensembles of several (say $n\geq 1$) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form $\exp[-\mathrm{Tr}(V_1)\times\ldots\times \mathrm{Tr}(V_k)]$ for certain noncommutative polynomials $V_1,\ldots,V_k\in \mathbb{C}_{\langle n \rangle}$ in the $n$ matrices. This article shows how the "algebra of functional renormalization" -- that is, the structure that makes the renormalization flow equation computable -- is derived from ribbon graphs, only by requiring the one-loop structure that such equation (due to Wetterich) is expected to have. Whenever it is possible to compute the renormalization flow in terms of $\mathrm U(N)$-invariants, the structure gained is the matrix algebra $M_n( \mathcal{A}_{n,N}, \star )$ with entries in $\mathcal{A}_{n,N}=(\mathbb{C}_{\langle n \rangle} \otimes \mathbb{C}_{\langle n \rangle} )\oplus( \mathbb{C}_{\langle n \rangle} \boxtimes \mathbb{C}_{\langle n \rangle})$, being $\mathbb{C}_{\langle n \rangle}$ the free algebra generated by the $n$ Hermitian matrices of size $N$ (the flowing random variables) with multiplication of homogeneous elements in $\mathcal{A}_{n,N}$ given, for each $P,Q,U,W\in\mathbb{C}_{\langle n \rangle}$, by \begin{align*}(U \otimes W) \star ( P\otimes Q) &= PU \otimes WQ \,, & (U\boxtimes W) \star ( P\otimes Q) &=U \boxtimes PWQ \,, \\(U \otimes W) \star ( P\boxtimes Q) &= WPU \boxtimes Q \,,\ & (U\boxtimes W) \star ( P\boxtimes Q) &= \mathrm{Tr} (WP) U\boxtimes Q \,,\end{align*} which, together with the condition $(\lambda U) \boxtimes W = U\boxtimes (\lambda W)$ for each complex $\lambda$, fully define the symbol $\boxtimes$.