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

November 04, 2021

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 $\boxtimes$.

Keywords:

Renormalization Group, Random Matrices, Noncommutative Algebra