  The nonperturbative renormalization group has been considered as a solid
framework to investigate fixed point and critical exponents for matrix and
tensor models, expected to correspond with the so-called double scaling limit.
In this paper, we focus on matrix models and address the question of the
compatibility between the approximations used to solve the exact
renormalization group equation and the modified Ward identities coming from the
regulator. We show in particular that standard local potential approximation
strongly violates the Ward identities, especially in the vicinity of the
interacting fixed point. Extending the theory space including derivative
couplings, we recover an interacting fixed point with a critical exponent not
so far from the exact result, but with a nonzero value for derivative
couplings, evoking a strong dependence concerning the regulator. Finally, we
consider a modified regulator, allowing to keep the flow not so far from the
ultralocal region and recover the results of the literature up to a slight
improvement.
