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

July 21, 2020

Random noncommutative geometry can be seen as a Euclidean path-integral quantization
approach to 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. As an
application of this formalism, we find the $\beta$-functions and identify the
fixed points in the large-$N$ limit of $2$-dimensional geometries in two
different signatures.

Keywords:

Renormalization Group, noncommutative geometry, matrix models, fuzzy geometry, quantum spacetime, large-N limit, free probability