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 ) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form for certain noncommutative polynomials in the 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 -invariants, the structure gained is the matrix algebra with entries in , being the free algebra generated by the Hermitian matrices of size (the flowing random variables) with multiplication of homogeneous elements in given, for each , by which, together with the condition for each complex , fully define the symbol .