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Random noncommutative geometry can be seen as a Euclidean path-integralapproach to the quantization of the theory defined by the Spectral Action innoncommutative geometry (NCG). With the aim of investigating phase transitionsin random NCG of arbitrary dimension, we study the non-perturbative FunctionalRenormalization Group for multimatrix models whose action consists ofnoncommutative polynomials in Hermitian and anti-Hermitian matrices. Suchstructure is dictated by the Spectral Action for the Dirac operator inBarrett's spectral triple formulation of fuzzy spaces.The presentmathematically rigorous treatment puts forward "coordinate-free" language thatmight be useful also elsewhere, all the more so because our approach holds forgeneral multimatrix models. The toolkit is a noncommutative calculus on thefree algebra that allows to describe the generator of the renormalization groupflow -- a noncommutative Laplacian introduced here -- in terms of Voiculescu'scyclic gradient and Rota-Sagan-Stein noncommutative derivative. We explore thealgebraic structure of the Functional Renormalization Group Equation and, as anapplication of this formalism, we find the $\beta$-functions, identify thefixed points in the large-$N$ limit and obtain the critical exponents of$2$-dimensional geometries in two different signatures.

On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra