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