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≥1) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form exp[−Tr(V1)×⋯×Tr(Vk)] for certain noncommutative polynomials V1,…,Vk∈C⟨n⟩ 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 U(N)-invariants, the structure gained is the matrix algebra M_n(A,⋆) with entries in A=(C⟨n⟩⊗C⟨n⟩)⊕(C⟨n⟩⊠C⟨n⟩), being C⟨n⟩ the free algebra generated by the n Hermitian matrices of size N (the flowing random variables) with multiplication of homogeneous elements in An,N given, for each P,Q,U,W∈C⟨n⟩, by (U⊗W)⋆(P⊗Q)=PU⊗WQ,(U⊠W)⋆(P⊗Q)=U⊠PWQ,(U⊗W)⋆(P⊠Q)=WPU⊠Q,(U⊠W)⋆(P⊠Q)=Tr(WP)U⊠Q, which, together with the condition (λU)⊠W=U⊠(λW) for each complex λ, fully define the symbol ⊠.