  Random matrices in the large N expansion and the so-called double scaling
limit can be used as toy models for quantum gravity: 2D quantum gravity coupled
to conformal matter. This has generated a tremendous expansion of random matrix
theory, tackled with increasingly sophisticated mathematical methods and number
of matrix models have been solved exactly. However, the somewhat paradoxical
situation is that either models can be solved exactly or little can be said.
Since the solved models display critical points and universal properties, it is
tempting to use renormalization group ideas to determine universal properties,
without solving models explicitly. Initiated by Br\'ezin and Zinn-Justin, the
approach has led to encouraging results, first for matrix integrals and then
quantum mechanics with matrices, but has not yet become a universal tool as
initially hoped. In particular, general quantum field theories with matrix
fields require more detailed investigations. To better understand some of the
encountered difficulties, we first apply analogous ideas to the simpler O(N)
symmetric vector models, models that can be solved quite generally in the large
N limit. Unlike other attempts, our method is a close extension of Br\'ezin and
Zinn-Justin. Discussing vector and matrix models with similar approximation
scheme, we notice that in all cases (vector and matrix integrals, vector and
matrix path integrals in the local approximation), at leading order,
non-trivial fixed points satisfy the same universal algebraic equation, and
this is the main result of this work. However, its precise meaning and role
have still to be better understood.
